Question

$f(x) = 5^{x}$
which of the followings is equal to $g(x)$?
a. $5^{(x - 3)} - 2$
b. $5^{(x - 2)} - 3$
c. $5^{(x - 2)} + 3$
d. $5^{(x + 3)} - 2$
$g(x) = ?$

Explanation:

Step1: Identify parent function key point

For $f(x)=5^x$, when $x=0$, $f(0)=5^0=1$. So key point $(0,1)$.

Step2: Find shifted key point

On $g(x)$, the corresponding key point is $(-3,1)$.

Step3: Determine horizontal shift rule

A shift from $x=0$ to $x=-3$ is $x \to x+3$, so $g(x)=5^{(x+3)} + k$.

Step4: Verify vertical shift

Check another point: $f(1)=5^1=5$ (point $(1,5)$). On $g(x)$, the matching point is $(-2,3)$. Substitute $x=-2$ into $5^{(x+3)} + k$: $5^{(-2+3)} + k = 5 + k = 3$. Solve for $k$: $k=3-5=-2$.

Step5: Form $g(x)$ equation

Combine shifts: $g(x)=5^{(x+3)}-2$.

Answer:

D. $5^{(x+3)}-2$