Find f(g(x)) and g (f(x)) and determine whether each pair of f...

Question

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=5x-9 and g(x) = (x+5)/9

Accepted Answer

Hello. Today we're going to be figuring out whether the given functions are in verses of each other. So the given F of X function is F of X is equal to three X cubed minus one. And the given G of X function is G of X is equal to the quantity X plus 1/3. That quantity raised to the 1/3 power. So how can we figure out whether they are in verses of each other? Well, in order to do that, we need to go ahead and first find the composite function F of G of X. Then we need to go ahead and find the composite function G of F of X. And if both of these are equal to each other and they produce the value of X then they are going to be in verses of each other. So let's go ahead and start with F of G of X. F of G of X means we're going to take our given G of X function and plug it directly into our F of X function. And doing so is going to give us three times the quantity that is going to be cubed X plus 1/3 raised to the one third power minus one. Now in order to simplify this portion here we need to go ahead and do exponential multiplication because we have an exponent raised to a power. So all we need to do is to go ahead and multiply one third and three together one third times three is going to give us one. So this simplifies to three times the quantity X plus 1/3 minus one. Now we can go ahead and simplify three times X plus 1/3 because the threes can cancel each other out. And what we are left with is X plus one minus one and one minus one is going to give us zero leaving us with just X. So we verify that F of G of X in verses of each other. But now we need to verify if G of F of X. R in verses now, just like before we're going to take our F of X function and plug it directly into our G of X function. Doing so is going to give us three X cubed minus one plus one. All over three. Everything raised to the one third power. Now we can start by simplifying the inside argument because negative one plus one is going to give us zero and that leaves us with three X cubed Over three raised to the 1/3 power. Now since we have a three in both the numerator and denominator of the inside argument, we can go ahead and cancel each other out and that's going to leave us with X cubed raised to the power of one third. And what we need to do now is to go ahead and apply exponential multiplication because we can just go ahead and multiply the exponents together three times a third is going to give us one. So this is going to leave us with just X. And that verifies that G of F of X. R. In verses of each other. So F of G of X is equal to G of F. Of X. And they both produce the value of X. So we verified that they are in verses of each other. And the answer to this problem is going to be a. So I hope this video helps you understanding how to verify whether functions are in verses of each other and I'll go ahead and see you all in the next video.

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=5x-9 and g(x) = (x+5)/9

Key Concepts

Function Composition

Inverse Functions

Algebraic Manipulation

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