Review Concepts for College Algebra Exam: Functions, Graphs, Transformations, and Equations

Functions and Their Properties

Definition and Types of Functions

In College Algebra, a function is a relation that assigns each element in the domain to exactly one element in the range. Functions can be represented in various forms, including equations, tables, graphs, and verbal descriptions.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) produced by the function.

• Piecewise Functions: Functions defined by different expressions over different intervals of the domain.

• Inverse Functions: Functions that reverse the effect of the original function, denoted as $f^{-1}(x)$.

Example: For $f(x) = x^2$, the domain is all real numbers, and the range is $[0, \\infty)$.

Graphs of Functions

Interpreting and Sketching Graphs

Graphs visually represent the behavior of functions. Key features include intercepts, intervals of increase/decrease, and symmetry.

• x-intercept: The point(s) where the graph crosses the x-axis ($f(x) = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

Example: The graph of $f(x) = x^2$ is a parabola opening upwards, symmetric about the y-axis.

Transformations of Functions

Types of Transformations

Transformations alter the position or shape of a function's graph. Common transformations include:

• Translations: Shifting the graph horizontally or vertically.

• Reflections: Flipping the graph over the x-axis or y-axis.

• Stretching/Compressing: Changing the graph's width or height.

Example: $f(x) = (x - 2)^2 + 3$ is the graph of $x^2$ shifted right by 2 units and up by 3 units.

Quadratic Functions

Standard and Vertex Form

Quadratic functions are polynomials of degree 2 and can be written in standard form $f(x) = ax^2 + bx + c$ or vertex form $f(x) = a(x-h)^2 + k$.

• Vertex: The highest or lowest point of the parabola, given by $(h, k)$ in vertex form.

• Axis of Symmetry: The vertical line $x = h$ passing through the vertex.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Example: For $f(x) = 2(x-1)^2 - 3$, the vertex is $(1, -3)$ and the parabola opens upward.

Piecewise Functions

Definition and Evaluation

A piecewise function is defined by different expressions depending on the input value.

• To evaluate, determine which interval the input belongs to and use the corresponding expression.

Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$

Function Operations and Composition

Combining Functions

Functions can be added, subtracted, multiplied, divided, or composed.

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(f \\cdot g)(x) = f(x)g(x)$

• Quotient: $(f / g)(x) = \frac{f(x)}{g(x)}$, $g(x) \\neq 0$

• Composition: $(f \\circ g)(x) = f(g(x))$

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \\circ g)(x) = f(g(x)) = (x + 1)^2$.

Solving Equations and Inequalities

Linear, Quadratic, and Rational Equations

Solving equations involves finding all values of the variable that make the equation true.

• Linear Equation: $ax + b = 0$

• Quadratic Equation: $ax^2 + bx + c = 0$

• Rational Equation: Involves fractions with polynomials in numerator and denominator.

• Inequality: Find the set of values that satisfy the inequality, often represented on a number line.

Example: Solve $x^2 - 3x + 2 = 0$ by factoring: $(x-1)(x-2) = 0$, so $x = 1$ or $x = 2$.

Tables and Function Evaluation

Using Tables to Represent Functions

Tables can be used to show input-output pairs for a function. This is useful for evaluating and comparing function values.

x	f(x)	g(x)
-2	2	3
-1	1	2
0	0	1
1	1	0
2	2	-1

Example: Use the table to find $f(1)$ and $g(1)$.

Exponential and Logarithmic Functions

Properties and Solving Equations

Exponential functions have the form $f(x) = a^x$, and logarithmic functions are their inverses, $f(x) = \\log_a(x)$.

• Exponential Equation: $a^{x} = b$

• Logarithmic Equation: $\\log_a(x) = b$

• Use properties of logarithms to solve equations, such as $\\log_a(xy) = \\log_a(x) + \\log_a(y)$.

Example: Solve $\\log_2(x) = 3$; $x = 2^3 = 8$.

Inverse Functions

Finding and Verifying Inverses

The inverse of a function reverses the input and output. To find the inverse, solve $y = f(x)$ for $x$ and interchange $x$ and $y$.

• Not all functions have inverses; the function must be one-to-one.

• Graphically, the inverse is a reflection over the line $y = x$.

Example: For $f(x) = 2x + 1$, solve $y = 2x + 1$ for $x$: $x = \\frac{y - 1}{2}$, so $f^{-1}(x) = \\frac{x - 1}{2}$.

Applications and Word Problems

Modeling with Functions

Functions are used to model real-world situations, such as population growth, cost analysis, and physical phenomena.

• Set up equations based on the context of the problem.

• Solve for unknowns using algebraic methods.

Example: If a box's volume is $V = lwh$, and you know two dimensions, solve for the third.

Summary Table: Function Transformations

Transformation	Equation	Effect on Graph
Vertical Shift	$f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal Shift	$f(x - h)$	Right if $h > 0$, left if $h < 0$
Reflection over x-axis	$-f(x)$	Flips graph upside down
Reflection over y-axis	$f(-x)$	Flips graph left-right
Vertical Stretch/Compression	$af(x)$	Stretches if $|a| > 1$, compresses if $0 < |a| < 1$

Practice Problems

Sample Questions

• Find the domain and range of $f(x) = \frac{1}{x-2}$.

• Graph $f(x) = |x|$ and $g(x) = -|x|$; describe the transformation.

• Solve the inequality $2x^2 - 3x < 5$.

• Find the inverse of $f(x) = 3x - 2$.

• Evaluate $f(g(x))$ for $f(x) = x^2$ and $g(x) = x + 1$.

Additional info: These notes synthesize the main concepts and sample problems from the provided exam review, covering functions, graphs, transformations, equations, and applications as relevant to College Algebra.