BackPrecalculus Chapter 1 & 2 Study Guide: Functions, Graphs, and Linear Equations
Study Guide - Smart Notes
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Q1. Tell whether or not the relation is a function: {(-6,7), (-2,-8), (4,-5), (4,6)}
Background
Topic: Functions and Relations
This question tests your understanding of what makes a relation a function, specifically whether each input (x-value) is paired with only one output (y-value).
Key Terms:
Relation: A set of ordered pairs.
Function: A relation where each input (x) has exactly one output (y).
Step-by-Step Guidance
List all the x-values in the relation: -6, -2, 4, 4.
Check if any x-value is repeated with a different y-value.
If an x-value is paired with more than one y-value, the relation is not a function.
Try solving on your own before revealing the answer!
Final Answer: No
The x-value 4 is paired with both -5 and 6, so the relation is not a function.
Q2. Determine the domain and range of the relation: {(-8, -7), (12, 7), (-4, -2), (-9, -1), (-2, 5)}
Background
Topic: Domain and Range of Relations
This question asks you to identify all possible input (domain) and output (range) values from a set of ordered pairs.
Key Terms:
Domain: The set of all x-values (inputs).
Range: The set of all y-values (outputs).
Step-by-Step Guidance
List all the x-values from the ordered pairs to find the domain.
List all the y-values from the ordered pairs to find the range.
Write the domain and range as sets, removing any duplicates.
Try solving on your own before revealing the answer!
Final Answer:
Domain: {12, -2, -9, -8, -4}; Range: {-7, 7, -2, -1, 5}
Q3. Given that $f(x) = x^2 - 5x + 1$, find $f(-4)$.
Background
Topic: Evaluating Functions
This question tests your ability to substitute a value for x into a function and simplify.
Key Formula:
$f(x) = x^2 - 5x + 1$
Step-by-Step Guidance
Substitute $x = -4$ into the function: $f(-4) = (-4)^2 - 5(-4) + 1$.
Calculate $(-4)^2$ and $-5(-4)$ separately.
Add the results together with the constant term.
Try solving on your own before revealing the answer!
Final Answer: 37
$f(-4) = 16 + 20 + 1 = 37$
Q4. Given that $f(x) = \frac{x}{11 - x}$, find $f\left(-\frac{4}{5}\right)$.
Background
Topic: Evaluating Rational Functions
This question asks you to substitute a fractional value into a rational function and simplify.
Key Formula:
$f(x) = \frac{x}{11 - x}$
Step-by-Step Guidance
Substitute $x = -\frac{4}{5}$ into the function: $f\left(-\frac{4}{5}\right) = \frac{-\frac{4}{5}}{11 - (-\frac{4}{5})}$.
Simplify the denominator: $11 - (-\frac{4}{5}) = 11 + \frac{4}{5}$.
Write $11$ as $\frac{55}{5}$ to combine with $\frac{4}{5}$.
Set up the final fraction for simplification.
Try solving on your own before revealing the answer!
Final Answer: $-\frac{4}{59}$
After simplifying, $f\left(-\frac{4}{5}\right) = -\frac{4}{59}$.
Q5. Find the domain of the function $f(x) = \frac{x}{x - 8}$.
Background
Topic: Domain of Rational Functions
This question tests your ability to determine the set of all real numbers for which the function is defined.
Key Terms:
Domain: All real numbers except where the denominator is zero.
Step-by-Step Guidance
Identify the denominator: $x - 8$.
Set the denominator not equal to zero: $x - 8 \neq 0$.
Solve for $x$ to find the value(s) to exclude from the domain.
Express the domain in set notation or interval notation, excluding the value found.
Try solving on your own before revealing the answer!
Final Answer:
Domain: $\{x|x \neq 8\}$, or $(-\infty, 8) \cup (8, \infty)$
Q6. Find the slope of the line containing the points (6, 5) and (8, 1).
Background
Topic: Slope of a Line
This question tests your ability to use the slope formula to find the rate of change between two points.
Key Formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Step-by-Step Guidance
Label the points: $(x_1, y_1) = (6, 5)$ and $(x_2, y_2) = (8, 1)$.
Substitute the values into the slope formula: $m = \frac{1 - 5}{8 - 6}$.
Simplify the numerator and denominator separately.
Try solving on your own before revealing the answer!
Final Answer: $-2$
The slope is $-2$.
Q7. Determine the slope, if it exists, of the graph of the linear equation $y = 4$.
Background
Topic: Slope of Horizontal and Vertical Lines
This question tests your understanding of the slope of special lines, such as horizontal lines.
Key Terms:
Horizontal line: Equation is $y = c$; slope is 0.
Vertical line: Equation is $x = c$; slope is undefined.
Step-by-Step Guidance
Recognize that $y = 4$ is a horizontal line.
Recall that the slope of a horizontal line is 0.
Try solving on your own before revealing the answer!
Final Answer: $m = 0$
The slope of a horizontal line is always 0.
Q8. Find the slope and the y-intercept of the line with the equation $y = 6 + \frac{2}{13}x$.
Background
Topic: Slope-Intercept Form
This question tests your ability to identify the slope and y-intercept from the equation of a line in slope-intercept form.
Key Formula:
Slope-intercept form: $y = mx + b$
$m$ is the slope, $b$ is the y-intercept.
Step-by-Step Guidance
Rewrite the equation as $y = \frac{2}{13}x + 6$ to match the standard form.
Identify the coefficient of $x$ as the slope $m$.
Identify the constant term as the y-intercept $b$.
Try solving on your own before revealing the answer!
Final Answer:
Slope: $\frac{2}{13}$; y-intercept: $(0, 6)$
Q9. To convert a temperature from degrees Celsius to degrees Fahrenheit, you multiply the temperature in degrees Celsius by 1.8 and then add 32 to the result. Find F as a linear function of c, and use this function to convert 33° C to °F.
Background
Topic: Linear Functions and Applications
This question tests your ability to write a linear function from a word description and use it to evaluate a specific value.
Key Formula:
$F = 1.8c + 32$
Step-by-Step Guidance
Write the function as described: $F = 1.8c + 32$.
Substitute $c = 33$ into the function: $F = 1.8 \times 33 + 32$.
Multiply $1.8 \times 33$.
Add 32 to the result from the previous step.
Try solving on your own before revealing the answer!
Final Answer: $91.4^ b0$F
$F = 1.8 \times 33 + 32 = 59.4 + 32 = 91.4$
