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 $x>0$. (D)</p> Signup and view all the answers

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

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

Flashcards

Find the equation of a line passing through (2, 2) with slope -3/7

The equation of a line passes through a given point (2, 2) and has a slope of -3/7. It is written in the slope-intercept form (y = mx + c): y = -3/7x + 20/7.

Finding the slope and y-intercept of 2x + 5y = 19

The slope-intercept form of the equation of a linear function is y = mx + c, where m represents the slope and c represents the y-intercept. In this case, the slope is -2/5 and the y-intercept is 19/5.

What is the domain and range of y = x² - 5?

For a polynomial function, the domain is all real numbers (-∞, ∞) since you can plug in any real number for x. The range is all real numbers greater than or equal to -5 ([-5, ∞]) because the graph is a parabola opening upwards with a vertex at -5.

Graphing a piecewise function

A piecewise-defined function has different rules for different parts of its domain. This function has two pieces: one rule for x ≤ 0 and another rule for x > 0. The graph consists of two lines: a horizontal line at y = -2 for x ≤ 0 and a line with slope 1 for x > 0.

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Finding the composite function g(f(x)) for f(x) = -6x + 4 and g(x) = 2x + 7

The expression g(f(x)) represents a function composition, where we first apply f(x) to obtain an output, and then apply g(x) to that output. We substitute (-6x + 4) for x in the expression for g(x) and simplify to obtain -12x + 15.

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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).