In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whethe...

Question

In Exercises 1-10, 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) = 3/(x-4) and g(x) = 3/x + 4

Accepted Answer

find out whether F and G are in verses of each other by finding F of G of X and G F F events. So we can tell if these are in verses of each other, if the compositions are equal to each other in other words, if F of G of X is equal to G of F of X. Let's go ahead and figure out the compositions. Let's first do F of G of X, which the same thing as saying F or G X inside apprentices F on the outside. So you think about this, you want F where G is our function, So seven over X plus 19. That function is going to be plugged in for X. So now everywhere in F where we have an X will replace it with seven over X plus 19. So let's look here, we have seven over X plus 19 is equal to seven divided by Then we have seven over X plus 19 minus 19. So let's go ahead and simplify this. Now I will get Seven defined by, we can combine our 19's 7 over X. We kinda have to think of this as two fractions on top of each other. Direct this seven has a one and a denominator To say 7/1 Over seven over X. Now, if you look here And each numerator, we have a seven, we can actually cross out the enumerators if they have the same number. So we cross that out and cross that seven out that goes to one And that other seven goes to what which means this will be 1/1 divided by one over X. We now get one of our one of her ex. Since we have a fraction on the bottom we can write it as a reciprocal with it on the top and multiplied both by X over one times X over one. I get X over one on the top over X over X on the bottom which just goes to X over one divided by one or X. So F composition is just equal to X. Let's see if our G composition gives the same thing. So you two are G G F. F. X. This is gonna be equal to G of seven over x months 19. This is equal to seven over seven Over X -19. And then we have a plus 19 on the outset. Let's go ahead and try and simplify this again. So you would get in the summer away 7/1 on top divided by seven over X minus 19. On the bottom Plus 19. On the outside the seven's will cancel meaning make it 1/1 over one over X minus 19 plus 19. As we get 1/1 over X - plus 19. We can simplify this. I'm multiplying by the reciprocal. This turns into X -19 on top plus 19. On the outside which means we just get X with the 19th canceling up. So they're both equal to X. Meaning that the answer to your problem is yes, F and G are in verses of each other. Okay. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 1-10, 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) = 3/(x-4) and g(x) = 3/x + 4

Key Concepts

Function Composition

Inverse Functions

Domain and Range

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