Select the correct answer.

Question

Which transformation of f(x) will produce the same graph as g(x)?

f(x) = 3x
g(x) = f(x) + 3

A.
h(x) = f(x − 1)
B.
h(x) = f(x) + 1
C.
h(x) = f(x + 1)
D.
h(x) = 2f(x)

Answer 1

To determine which transformation of $f(x)$ will produce the same graph as $g(x)$, we need to understand what transformation has been applied to $f(x)$.

Given functions:
$$f(x)=3x$$
$$g(x)=f(x)+3$$

Rewriting $g(x)$:
$$g(x)=3x+3$$

We compare $g(x)$ to $f(x)$ and see that $g(x)$ is simply $f(x)$ shifted upward by 3 units. Therefore, we need to find a transformation that results in shifting $f(x)$ up by 3 units.

Examining the options:
A. $h(x)=f(x-1)$ - This represents a horizontal shift to the right by 1 unit.
B. $h(x)=f(x)+1$ - This represents a vertical shift up by 1 unit.
C. $h(x)=f(x+1)$ - This represents a horizontal shift to the left by 1 unit.
D. $h(x)=2f(x)$ - This represents a vertical stretch by a factor of 2.

The correct transformation to shift $f(x)$ up by 3 units is:
$$h(x)=f(x)+3$$

Since $g(x)=f(x)+3$, it appears the question might have a misprint because none of the provided options exactly match $h(x)=f(x)+3$. However, under the assumption the problem intends to find h(x) = f(x) + 3, the correct answer should ideally reflect:

$$h(x)=f(x)+3$$

Though it's not directly provided in the options, for the purpose of selecting the correct answer given what's available:
None of the provided options exactly match $g(x)=f(x)+3$, suggesting a necessary correction in either the provided transformations or reviewing other integrally intended context.