Question

find an equation for g(x) when g(x) = f(-x). graph both functions on the same coordinate grid.
$f(x) = \frac{x}{3} + 3$
$g(x) = -\frac{x}{3} + 3$
(use integers or fractions for any numbers in the expression.)
determine the effects on the graph of f(x) when g(x) = f(-x). select the correct choice below and, if necessary,
\\(\bigcirc\\) a. reflection across the x-axis, y-intercept changed to
\\(\bigcirc\\) b. reflection across the y-axis, y-intercept changed to
\\(\bigcirc\\) c. reflection across the x-axis, same y-intercept
\\(\bigcirc\\) d. reflection across the y-axis, same y-intercept

Explanation:

Step1: Substitute -x into f(x)

Given \( f(x) = \frac{x}{3} + 3 \), replace \( x \) with \( -x \):
\( g(x) = f(-x) = \frac{-x}{3} + 3 = -\frac{x}{3} + 3 \)

Step2: Analyze the transformation

\( g(x) = f(-x) \) means reflection over y-axis. Check y-intercepts:
For \( f(x) \), y-intercept is \( f(0) = 3 \); for \( g(x) \), \( g(0) = -\frac{0}{3} + 3 = 3 \). Same y-intercept.

Answer:

Equation of \( g(x) \): \( -\frac{x}{3} + 3 \)
Correct option: D. Reflection across the y-axis, same y-intercept