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A function $g(x)$ increases by $4$ over every unit interval in $x$ and $g(0) = 0$ . Which could be a function rule for $g(x)$ ? $\newline$ Choices: $\newline$ (A) $g(x) = -4x$ $\newline$ (B) $g(x) = 4^x$ $\newline$ (C) $g(x) = 4x$ $\newline$ (D) $g(x)$ = $-x$ - $4$

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Transformations of absolute value functions

Full solution

Q. A function

g(x)

increases by

4

over every unit interval in

x

and

g(0) = 0

. Which could be a function rule for

g(x)

?

\newline

Choices:

\newline

(A)

g(x) = -4x

\newline

(B)

g(x) = 4^x

\newline

(C)

g(x) = 4x

\newline

(D)

g(x)

=

-x

-

4

Start at origin: Since $g(0) = 0$ , the function starts at the origin. This eliminates any function with a non-zero $y$ -intercept.
Determine linear function: The function increases by $4$ for each increase of $1$ in $x$ , so the slope must be $4$ . This means the function is linear and of the form $g(x) = 4x + b$ .
Substitute $x=0$ : Since $g(0) = 0$ , we can substitute $x = 0$ into the equation to find $b$ : $g(0) = 4(0) + b$ , which simplifies to $0 = b$ .
Find function rule: Now we know that $b = 0$ , so the function rule is $g(x) = 4x$ .
Match function rule: Looking at the choices, (C) $g(x) = 4x$ matches our function rule.

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Question

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What is the domain of this function?

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\newline

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