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Algebra Class: Functions and Equations

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Questions and Answers

A line passes through the point (2, 2) and has a slope of $-\frac{3}{7}$. Which of the following is the equation of this line?

$y = -\frac{3}{7}x - \frac{20}{7}$

$y = \frac{3}{7}x - \frac{20}{7}$

$y = -\frac{3}{7}x + \frac{20}{7}$ (correct)

$y = \frac{3}{7}x + \frac{20}{7}$

What are the slope and y-intercept of the line given by the equation $2x + 5y = 19$?

Slope: $\frac{2}{5}$, y-intercept: $-\frac{19}{5}$

Slope: $\frac{5}{2}$, y-intercept: $\frac{19}{5}$

Slope: $-\frac{2}{5}$, y-intercept: $\frac{19}{5}$ (correct)

Slope: $-\frac{5}{2}$, y-intercept: $-\frac{19}{5}$

What is the range of the function $y = x^2 - 5$?

$(-\infty, \infty)$

$\left[-5, \infty\right)$ (correct)

$\left(0, \infty\right)$

$\left(-\infty, -5\right]$

Consider the piecewise-defined function: $f(x) = \begin{cases} -2, & x \leq 0 \ x + 1, & x > 0 \end{cases}$. Which of the following statements best describes the graph of this function?

<p>The graph consists of two line segments, one horizontal at $y=-2$ for $x\leq 0$, and another with slope of 1 and y-intercept of 1 for $x>0$.</p> Signup and view all the answers

If $f(x) = -6x + 4$ and $g(x) = 2x + 7$, what is $g(f(x))$?

<p>$-12x + 15$</p> Signup and view all the answers

Study Notes

Equation of a Line

Given a point (2, 2) and slope -3, the equation is y = -3x + 8
The slope of the line 2x + 5y = 19 is -2/5
The y-intercept is 19/5

Domain and Range

For the function y = x² - 5, the domain is all real numbers
The range is y ≥ -5

Piecewise Function

The function f(x) is defined as x + 1 if x ≤ 0 and x if x > 0.
The graph of this function has a point at (0, 1). When x=0, and when x=2 the function goes through point (2, 2).

Function Composition

Given f(x) = -6x + 4 and g(x) = 2x + 7, g(f(x)) = -12x + 15

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Description

This quiz covers key concepts in algebra, including the equations of lines, domain and range, piecewise functions, and function composition. Test your understanding of these fundamental topics with practical examples and problem-solving scenarios.