Answer: Choice C
f(x) = 13x-7 and g(x) = (x+7)/13
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Step-by-step explanation:
Let's go through the answer choices to see which pairs are inverses or not.
The process of finding an inverse follows these steps
- Replace f(x) with y
- Swap x and y
- Solve for y to determine the inverse g(x)
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Choice A
f(x) = (x/3) + 10
y = (x/3) + 10 .... f(x) replaced with y
x = (y/3) + 10 .... x and y swapped; solve for y from here
x-10 = y/3
y = 3(x-10)
y = 3x-30
g(x) = 3x-30
This doesn't match the g(x) = 3x-10 listed. We can rule out choice A.
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Choice B
f(x) = cube root of 4x
y = cube root of 4x
x = cube root of 4y
x^3 = 4y
4y = x^3
y = (1/4)x^3
g(x) = (1/4)x^3
This is close to what is listed, but the 4 is not being cubed. We can rule out choice B as well.
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Choice C
f(x) = 13x-7
y = 13x-7
x = 13y-7
x+7 = 13y
13y = x+7
y = (x+7)/13
g(x) = (x+7)/13
This is a perfect match to what choice C is saying. Therefore, choice C is the final answer. If you applied this process in reverse, you'd find that g(x) leads to the inverse f(x).
Further confirmation would be to show that f(g(x)) = x and g(f(x)) = x for all real numbers x. I'll let you do this step.
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For the sake of completion, let's check choice D
f(x) = (9/x) - 5
y = (9/x) - 5
x = (9/y) - 5
x+5 = 9/y
y(x+5) = 9
y = 9/(x+5)
g(x) = 9/(x+5)
Choice D lists (x+5)/9 instead of 9/(x+5), so we can rule out choice D.