Question

a. graph ( g(x) = fleft( \frac{1}{3}x
ight) )
b. complete the sentences to describe the graph of ( g ) compared to the graph of ( f ).
the graph of ( g ) is a (vertical/horizontal) __ scaling of the graph of ( f ) with scale factor _. the graph of ( g ) is the graph of ( f ) (compressed/stretched) (vertically/horizontally) __

c. write the domains and the ranges of ( f ) and ( g ).

function	domain	range
( g )

Explanation:

Step1: Identify key points of $f(x)$

From the graph, key points of $f(x)$ are: $(-8, -2)$, $(-4, 0)$, $(2, 6)$, $(9, 5)$

Step2: Transform points for $g(x)=f(\frac{1}{3}x)$

For horizontal stretch by factor 3: solve $\frac{1}{3}x = x_f \implies x = 3x_f$.
New points:
$x=3(-8)=-24, y=-2$ → $(-24, -2)$
$x=3(-4)=-12, y=0$ → $(-12, 0)$
$x=3(2)=6, y=6$ → $(6, 6)$
$x=3(9)=27, y=5$ → $(27, 5)$

Step3: Describe transformation type

$g(x)=f(kx)$ with $|k|<1$ is horizontal stretch.

Step4: Find domain/range of $f(x)$

Domain of $f$: $-8 \leq x \leq 9$
Range of $f$: $-2 \leq y \leq 6$

Step5: Find domain/range of $g(x)$

Domain of $g$: $3(-8) \leq x \leq 3(9) \implies -24 \leq x \leq 27$
Range of $g$ (unchanged): $-2 \leq y \leq 6$

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Answer:

Part a

Plot the transformed points $(-24, -2)$, $(-12, 0)$, $(6, 6)$, $(27, 5)$ and connect them to match the shape of $f(x)$, stretched horizontally.

Part b

The graph of $g$ is a horizontal scaling of the graph of $f$ with scale factor $\boldsymbol{3}$. The graph of $g$ is the graph of $f$ stretched horizontally.

Part c

Function	Domain	Range
$g$	$-24 \leq x \leq 27$	$-2 \leq y \leq 6$