Common Functions: Videos & Practice Problems

Understanding functions is crucial in mathematics, particularly common types like constant, identity, square, cube, square root, and cube root functions. Each function has specific domains and ranges, such as the constant function having a range of a single value, while quadratic functions exhibit parabolic shapes with unique vertices and intercepts. Additionally, mastering slope-intercept form, $y=\mathrm{mx}+b$, is essential for writing linear equations. Recognizing these concepts enhances problem-solving skills and prepares students for advanced topics in algebra and calculus.

Graphs of Common Functions

Understanding common functions is essential for mastering mathematical concepts. This summary explores several key functions, their definitions, and their characteristics, including domain and range.

The constant function is defined as f(x) = c, where c is any constant number. For example, if f(x) = 2, the graph is a horizontal line at y = 2. The domain of a constant function is all real numbers, represented as (−∞, +∞), since any value of x can be input. However, the range is limited to the constant value, so in this case, it is simply {2}.

The identity function is expressed as f(x) = x. This function outputs the same value as the input, meaning if you input -1, the output is -1, and if you input 50, the output is 50. Both the domain and range for the identity function are all real numbers, (−∞, +∞).

Next, the square function is defined as f(x) = x², producing a parabolic graph. The domain is all real numbers, (−∞, +∞), as the curve extends infinitely in both directions. However, the range is restricted to non-negative values, [0, +∞), since the output cannot be negative.

The cube function is given by f(x) = x³. This function includes all real numbers in both the domain and range, (−∞, +∞), as the graph extends infinitely in all directions.

The square root function is represented as f(x) = √x. This function has a restricted domain of non-negative values, [0, +∞), because square roots of negative numbers are not defined in the real number system. The range is also [0, +∞), as the output cannot be negative.

Lastly, the cube root function is defined as f(x) = ∛x. Similar to the cube function, the domain and range for the cube root function encompass all real numbers, (−∞, +∞), allowing for both negative and positive inputs and outputs.

These functions form the foundation for more complex mathematical concepts and are frequently encountered in various mathematical contexts. Familiarity with their properties, including domain and range, is crucial for success in future studies.

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing and analyzing lines. The slope-intercept form is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) denotes the y-intercept.

The slope, \(m\), is calculated using the formula for rise over run, which can be expressed as \(m = \frac{\Delta y}{\Delta x}\). This means you determine how much the line rises (the change in y) for a given run (the change in x) between two points on the line. For example, if the rise is 2 and the run is 1, the slope would be \(m = 2\).

The y-intercept, \(b\), is the point where the line crosses the y-axis, which occurs when \(x = 0\). For instance, if the line crosses the y-axis at the point (0, 3), then \(m\) and \(b\) into the equation. For example, if the slope is 2 and the y-intercept is 3, the equation becomes \(y = 2x + 3\).

In another example, if a line crosses the y-axis at -3 and has a slope of 1, the equation can be written as \(y = 1x - 3\), which simplifies to \(y = x - 3\).

By mastering the slope-intercept form, you can easily describe and graph linear equations, making it a fundamental skill in algebra.

In the graph shown, identify the y–intercept & slope. Write the equation of this line in Slope-Intercept form.

• A. $y=\frac23x+1$
• B. $y=-\frac23x+1$
• C. $y=-2x+1$
• D. $y=x+2$

Answer: (B)

Properties of Parabolas

Quadratic functions are a specific type of polynomial function characterized by their degree of 2. The standard form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real numbers, and \( a \neq 0 \). The graph of any quadratic function takes the shape of a parabola, which can open either upward or downward depending on the sign of \( a \).

The vertex of a parabola is a crucial feature, representing either the highest or lowest point on the graph. If the parabola opens upward, the vertex is a minimum point; if it opens downward, the vertex is a maximum point. The vertex is denoted as an ordered pair, such as \( (h, k) \), where \( h \) and \( k \) are the x and y coordinates, respectively.

Another important aspect of quadratic functions is the x-intercepts, which are the points where the graph crosses the x-axis. A quadratic function can have either one or two x-intercepts, but never more. The y-intercept, on the other hand, is where the graph intersects the y-axis, and it can be found by evaluating \( f(0) \).

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, always passing through the vertex. It can be expressed as \( x = h \), where \( h \) is the x-coordinate of the vertex.

When analyzing the domain and range of quadratic functions, the domain is always all real numbers, represented as \( (-\infty, \infty) \). The range, however, depends on whether the vertex is a minimum or maximum. For a minimum point, the range extends from the vertex's y-coordinate to infinity, while for a maximum point, it extends from negative infinity to the vertex's y-coordinate.

Additionally, understanding the intervals of increase and decrease is essential. A parabola increases on the interval from negative infinity to the vertex and decreases from the vertex to positive infinity. This behavior is determined by the vertex's position on the graph.

Quadratic functions can also be expressed in vertex form, which is particularly useful for graphing. The vertex form is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. This form allows for easier identification of the vertex and the direction in which the parabola opens.

Identify the ordered pair of the vertex of the parabola. State whether it is a minimum or maximum.

• A. $\left(-3,-2\right);$ maximum
• B. $\left(-3,-2\right);$ minimum
• C. $\left(-1,2\right);$ maximum
• D. $\left(-1,2\right);$ minimum

Answer: (B)

Where is the axis of symmetry located on the given parabola? 

• A. $x=3$
• B. $x=0$
• C. $x=1$
• D. $x=4$

Answer: (A)

Introduction to Polynomial Functions

Polynomial functions are a broad category of mathematical expressions characterized by their use of positive whole number exponents. A polynomial function can be expressed in the form f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where a_n is the leading coefficient, n is the degree of the polynomial, and the terms are arranged in descending order of power. For example, in the polynomial f(x) = 5x³ - x² - 6x + 4, the degree is 3, and the leading coefficient is 5.

To determine if a given expression is a polynomial function, it is essential to check that all exponents are positive whole numbers. For instance, f(x) = 2x^1/2 + 3 is not a polynomial function due to the fractional exponent. However, f(x) = -\frac{2}{3}x⁴ + 4 is valid since it contains a fraction only as a coefficient, not in the exponent.

The graphs of polynomial functions exhibit two key characteristics: they are smooth and continuous. This means that the graph will not have any corners or breaks. For example, a quadratic function, which is a specific type of polynomial function, maintains this smooth and continuous nature. In contrast, graphs that display sharp corners or breaks do not represent polynomial functions.

Additionally, the domain of polynomial functions is always all real numbers, extending from negative infinity to positive infinity. This property is consistent across all polynomial functions, including quadratics.

Understanding these foundational aspects of polynomial functions is crucial for further exploration of their properties and applications in mathematics.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

• A. Polynomial with $n=3,a_{n}=4$
• B. Polynomial with $n=4,a_{n}=3$
• C. Polynomial with $n=-1,a_{n}=\frac12$
• D. Not a polynomial function.

Answer: (D)

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=2+x$

• A. Polynomial with $n=1,a_{n}=2$
• B. Polynomial with $n=0,a_{n}=1$
• C. Polynomial with $n=1,a_{n}=1$
• D. Not a polynomial function.

Answer: (C)

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

• A. Polynomial with $n=3,a_{n}=2$
• B. Polynomial with $n=2,a_{n}=3$
• C. Polynomial with $n=2,a_{n}=2$
• D. Not a polynomial function.

Answer: (B)

Intro to Rational Functions

Rational functions are formed by placing one polynomial over another in a fraction, represented as \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials of any degree. Understanding rational functions involves recognizing that the denominator cannot equal zero, which leads to domain restrictions. For example, if the denominator is \( x - 1 \), then \( x \) cannot equal 1, as this would make the denominator zero. This restriction is crucial for defining the domain of the function, which can be expressed in set notation as \( \{ x \, | \, x \neq 1 \} \).

To find the domain of a rational function, set the denominator equal to zero and solve for \( x \). For instance, for the function \( \frac{3}{3x - 12} \), setting the denominator to zero gives \( 3x - 12 = 0 \). Solving this results in \( x = 4 \), indicating that the domain is all real numbers except \( x = 4 \). Similarly, for \( f(x) = \frac{x + 5}{x^2 - 25} \), setting the denominator \( x^2 - 25 = 0 \) leads to \( x = 5 \) and \( x = -5 \), resulting in the domain restriction \( x \neq 5 \) and \( x \neq -5 \).

Another important aspect of rational functions is simplifying them to their lowest terms. This involves factoring both the numerator and the denominator and canceling any common factors. For example, in the function \( \frac{3}{3x + 12} \), the denominator can be factored to \( 3(x + 4) \), allowing the common factor of 3 to be canceled, simplifying the function to \( \frac{1}{x + 4} \).

In the case of \( f(x) = \frac{x + 5}{x^2 - 25} \), the denominator factors as \( (x + 5)(x - 5) \). Canceling the common factor \( x + 5 \) results in \( \frac{1}{x - 5} \). However, it is essential to determine the domain before simplifying, as simplifying may obscure additional restrictions. Thus, the domain must always be established first to ensure all restrictions are accounted for.

In summary, when working with rational functions, it is vital to identify domain restrictions by setting the denominator to zero, factor and simplify the functions carefully, and always confirm the domain before any simplification to avoid missing critical restrictions.

Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{x^2+9}{x-3}$

• A. {x∣x≠0}, $f\left(x\right)=\frac{1}{x-3}$
• B. {x∣x≠3}, $f\left(x\right)=\frac{x^2+9}{x-3}$
• C. {x∣x≠−3}, $f\left(x\right)=\frac{x^2+9}{x-3}$
• D. {x∣x≠3}, $f\left(x\right)=x+3$

Answer: (B)

Find the domain of the rational function. Then, write it in lowest terms.
$f\left(x\right)=\frac{6x^5}{2x^2-8}$

• A. ${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$
• B. ${\left\lbrace x|x\ne2,-2\right\rbrace},f(x)=\frac{6x^5}{2x^2-8}$
• C. ${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-4}$
• D. ${\left\lbrace x|x\ne2\right\rbrace},f(x)=\frac{3x^5}{x^2-8}$

Answer: (A)

Exponential Functions

Exponential functions are a fundamental concept in mathematics, distinct from polynomial functions. A basic example of an exponential function is f(x) = 2^x, where the base (2) is a constant and the exponent (x) is a variable. Understanding the characteristics of the base and exponent is crucial. The base must be a positive constant and cannot equal 1, while the exponent must contain a variable.

For instance, consider the function f(x) = (2/3)^x. Here, the base is 2/3, which is positive and not equal to 1, confirming that this is indeed an exponential function. The exponent is simply x, which is a variable. Conversely, the function f(y) = 1^y does not qualify as an exponential function because the base is 1, which violates the criteria.

Another example is f(x) = 10^(x+1). In this case, the base is 10, which is positive and not equal to 1, while the exponent is x + 1, containing the variable x. Thus, this function is also an exponential function.

Evaluating exponential functions involves substituting values for the variable. For example, to evaluate f(x) = 2^x at x = 4, we compute f(4) = 2^4 = 16. When evaluating at a negative exponent, such as x = -3, we find f(-3) = 2^{-3} = 1/(2^3) = 1/8.

For non-integer values, like x = 3.14, a calculator is often necessary. Inputting 2^3.14 using the caret key on a calculator yields approximately 8.815. Similarly, for larger exponents, such as x = 12, calculating f(12) = 2^{12} results in 4096.

In summary, exponential functions are characterized by a constant positive base and a variable exponent. Evaluating these functions requires substituting values for the exponent and can often be facilitated by using a calculator for complex calculations.

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$.
$f\left(x\right)=\left(-2\right)^{x}$

• A. Exponential function, $f\left(4\right)=16$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function

Answer: (C)

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$ .
$f\left(x\right)=3\left(1.5\right)^{x}$

• A. Exponential function, $f\left(4\right)=410.06$
• B. Exponential function, $f\left(4\right)=15.19$
• C. Not an exponential function

Answer: (B)

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$ .
$f\left(x\right)=\left(\frac12\right)^{x}$

• A. Exponential function, $f\left(4\right)=\frac{1}{16}$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function

Answer: (A)

Logarithms Introduction

When solving polynomial equations, such as \( x^3 = 216 \), the goal is to isolate \( x \) by performing the reverse operation. In this case, taking the cube root of both sides allows us to find that \( x = \sqrt[3]{216} \). However, when the variable \( x \) is in the exponent, as in \( 2^x = 8 \), we need to determine how many times 2 must be multiplied by itself to equal 8, leading us to conclude that \( x = 3 \).

For more complex equations like \( 2^x = 216 \), instead of multiplying 2 repeatedly, we can utilize logarithms. The logarithm is the inverse operation of exponentiation, allowing us to isolate \( x \) effectively. To do this, we take the logarithm of both sides, specifically using the same base as the exponential. Thus, we write \( \log_2(2^x) = \log_2(216) \). Since the logarithm and the exponential share the same base, they cancel out, resulting in \( x = \log_2(216) \).

The expression \( \log_2(216) \) represents the power to which the base 2 must be raised to yield 216. This logarithmic form is equivalent to the original exponential equation \( 2^x = 216 \). Understanding how to convert between these forms is crucial. For instance, converting \( 3^x = 81 \) into logarithmic form involves starting with the base 3, leading to \( \log_3(81) = x \). Conversely, if we start with \( x = \log_4(64) \), we can express it in exponential form as \( 4^x = 64 \).

To further illustrate this process, consider \( x = \log_5(800) \). The conversion to exponential form begins with the base 5, resulting in \( 5^x = 800 \). Similarly, for \( \log_2(16) = 4 \), we convert it to \( 2^4 = 16 \), confirming the accuracy of our conversion. Lastly, for \( 10^x = 45100 \), we can express it as \( x = \log_{10}(45100) \), which is often referred to as the common logarithm and can simply be written as \( \log(45100) \).

Understanding these conversions between logarithmic and exponential forms is essential for solving equations involving exponents and for applying logarithmic properties in various mathematical contexts.

Graphs of Logarithmic Functions

Graphing logarithmic functions can initially seem daunting, but it closely mirrors the process of graphing exponential functions due to their inverse relationship. For instance, when working with the logarithmic function log2(x), we can derive ordered pairs by substituting values for x. Starting with x = 1, we find that log2(1) = 0, giving us the point (1, 0). Next, for x = 2, log2(2) = 1, resulting in the point (2, 1).

These points can be compared to the corresponding exponential function f(x) = 2x, where the points (0, 1) and (1, 2) can be flipped to yield the points for the logarithmic function. This flipping of coordinates is a fundamental property of inverse functions, allowing us to easily generate the necessary points for our logarithmic graph.

As we plot these points, we observe that the graph approaches the y-axis but never touches it, indicating a vertical asymptote at x = 0. This behavior is consistent across all logarithmic functions, where the domain is all positive real numbers ((0, ∞)) and the range is all real numbers ((−∞, ∞)).

To visualize the relationship between the graphs of exponential and logarithmic functions, one can imagine reflecting the exponential graph over the line y = x. This reflection illustrates how the two functions are inverses of each other. Additionally, the shapes of these graphs can be remembered by associating the exponential function with a lowercase "e" and the logarithmic function with a lowercase "r," where extending these letters resembles the respective graph shapes.

When considering logarithmic functions with different bases, the behavior of the graph remains consistent with that of exponential functions. For bases greater than 1, the logarithmic graph will be increasing, while for bases between 0 and 1, the graph will be decreasing. This understanding of base influence is crucial for accurately graphing logarithmic functions.

In summary, mastering the graphing of logarithmic functions involves recognizing their inverse relationship with exponential functions, utilizing coordinate flipping, and understanding the implications of different bases on the graph's behavior. With practice, graphing logarithmic functions will become a straightforward extension of the skills developed while working with exponential functions.

Graphing Logarithmic Functions Example 1

In this exploration of functions, we focus on two specific mathematical expressions: \( f(x) = 3^x \) and \( g(x) = \log_3(x) \). These functions are inverses of each other, as the logarithmic function serves as the inverse of the exponential function. Understanding their graphs provides insight into their behaviors and relationships.

To graph \( f(x) = 3^x \), we begin by calculating key points. For \( x = 0 \), we find \( f(0) = 3^0 = 1 \). For \( x = 1 \), \( f(1) = 3^1 = 3 \), and for \( x = 2 \), \( f(2) = 3^2 = 9 \). These points (0, 1), (1, 3), and (2, 9) illustrate the rapid growth of the function. For negative values, we calculate \( f(-1) = 3^{-1} = \frac{1}{3} \) and \( f(-2) = 3^{-2} = \frac{1}{9} \), showing that as \( x \) decreases, \( f(x) \) approaches zero but never touches the x-axis, indicating a horizontal asymptote at \( y = 0 \).

Next, we graph \( g(x) = \log_3(x) \). To find points for this function, we can swap the x and y values from the graph of \( f(x) \). The point (1, 3) becomes (3, 1), (0, 1) becomes (1, 0), and (2, 9) becomes (9, 2). The logarithmic function approaches the y-axis but never crosses it, indicating a vertical asymptote at \( x = 0 \). The points plotted for \( g(x) \) include (1, 0), (3, 1), and (9, 2), demonstrating its growth as \( x \) increases.

When analyzing the domains and ranges of these functions, we find that the domain of \( f(x) = 3^x \) is all real numbers, \( (-\infty, \infty) \), while its range is \( (0, \infty) \) due to the horizontal asymptote. Conversely, the domain of \( g(x) = \log_3(x) \) is \( (0, \infty) \), and its range is all real numbers, \( (-\infty, \infty) \). This relationship between the domain and range of inverse functions reinforces the concept that they are reflections across the line \( y = x \).

In summary, the graphs of \( f(x) = 3^x \) and \( g(x) = \log_3(x) \) illustrate the fundamental properties of exponential and logarithmic functions, including their growth behaviors, asymptotic characteristics, and the relationship between their domains and ranges.

Here’s what students ask on this topic:

What is the domain and range of a constant function?

A constant function is defined as $f\left(x\right)=c$, where $c$ is a constant. The domain of a constant function is all real numbers, which means you can input any value for $x$. In interval notation, this is expressed as $(-\infty ,\infty )$. The range, however, is limited to the single value $c$, since the output of the function is always the same regardless of the input. Therefore, the range is $\left\{c\right\}$.

How do you find the slope and y-intercept of a line in slope-intercept form?

The slope-intercept form of a line is given by the equation $y=mx+b$, where $m$ represents the slope and $b$ is the y-intercept. To find the slope, identify the coefficient of $x$ in the equation, which is $m$. The slope indicates the steepness and direction of the line. The y-intercept is the constant term $b$, which is the point where the line crosses the y-axis. This point is represented as $(0,b)$.

What are the characteristics of a quadratic function's graph?

A quadratic function is expressed as $f\left(x\right)=ax²+bx+c$. Its graph is a parabola, which can open upwards or downwards depending on the sign of $a$. Key features include the vertex, which is the highest or lowest point of the parabola, and the axis of symmetry, a vertical line that passes through the vertex. The x-intercepts are the points where the graph crosses the x-axis, and the y-intercept is where it crosses the y-axis. The domain of a quadratic function is all real numbers, while the range depends on the direction of the parabola.

How do you determine the domain of a rational function?

A rational function is expressed as $f\left(x\right)=\frac{p}{\left(}$, where $p\left(x\right)$ and $q\left(x\right)$ are polynomials. The domain of a rational function includes all real numbers except those that make the denominator zero. To find these restrictions, set the denominator $q\left(x\right)$ equal to zero and solve for $x$. The solutions are the values that are excluded from the domain. In interval notation, express the domain by excluding these values.

What is the relationship between exponential and logarithmic functions?

Exponential and logarithmic functions are inverses of each other. An exponential function is of the form $f\left(x\right)=b^x$, where $b$ is a constant base. A logarithmic function is expressed as $g\left(x\right)=\log \_b\left(x\right)$. The logarithm gives the power to which the base must be raised to obtain a given number. Graphically, the exponential function is a reflection of the logarithmic function over the line $y=x$. The domain of an exponential function is all real numbers, while its range is positive real numbers. Conversely, the domain of a logarithmic function is positive real numbers, and its range is all real numbers.