Question

the functions $f(x)$ and $g(x)$ are defined by their equations below.
$f(x) = 2x - 6$
$g(x) = \frac{1}{3}x + 2$
compare the key features of $f(x)$ and $g(x)$ by filling in the table below.
show your work here

	$f(x)$	$g(x)$
$x$-intercept	enter your answer	enter your answer
$y$-intercept	enter your answer	enter your answer

Explanation:

Step1: Recall slope-intercept form

The slope-intercept form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

Step2: Find slope and intercepts for \( f(x) \)

For \( f(x) = 2x - 6 \):

• Slope (\( m \)): The coefficient of \( x \) is \( 2 \), so slope is \( 2 \).
• \( y \)-intercept (\( b \)): When \( x = 0 \), \( f(0) = 2(0) - 6 = -6 \), so \( y \)-intercept is \( -6 \).
• \( x \)-intercept: Set \( f(x) = 0 \), so \( 2x - 6 = 0 \). Solving for \( x \): \( 2x = 6 \) \( \Rightarrow x = 3 \).

Step3: Find slope and intercepts for \( g(x) \)

For \( g(x) = \frac{1}{3}x + 2 \):

• Slope (\( m \)): The coefficient of \( x \) is \( \frac{1}{3} \), so slope is \( \frac{1}{3} \).
• \( y \)-intercept (\( b \)): When \( x = 0 \), \( g(0) = \frac{1}{3}(0) + 2 = 2 \), so \( y \)-intercept is \( 2 \).
• \( x \)-intercept: Set \( g(x) = 0 \), so \( \frac{1}{3}x + 2 = 0 \). Solving for \( x \): \( \frac{1}{3}x = -2 \) \( \Rightarrow x = -6 \).

Answer:

Feature	\( f(x) \)	\( g(x) \)
\( x \)-intercept	\( 3 \)	\( -6 \)
\( y \)-intercept	\( -6 \)	\( 2 \)