Question

Use function composition to verify f(x)=-3x+5 and g(x)=\frac{x-5}{-3} are inverses. When typing your answer if you have an exponent then use the carrot key ^ by pressing SHIFT and 6. Type your simplified answers in descending powers of x an do not include any spaces between your characters.Type your answer for this composition without simplifying:the numerator of g(f(x))=Answerthe denominator g(f(x))=AnswerNow simplify the composition, are f(x) and g(x) inverses? Answer

Answer 1

We are given the following functions:

`\begin{gathered} f(x)=-3x+5 \\ g(x)=(x-5)/(-3) \end{gathered}`

We are asked to determine the composite function:

`g(f(x))`

To do that we will replace as the value of "x" in g(x) the function f(x), like this:

`g(f(x))=((-3x+5)-5)/(-3)`

Therefore, the numerator of the composite function is:

`\text{ numerator g(f(x))=-3x+5-5}`

And the denominator is:

`\text{ denominator g(f(x))=-3}`

Now we simplify the fraction, first by canceling out the 5:

`g(f(x))=(-3x)/(-3)`

Now we cancel out the -3:

`g(f(x))=x`

Since the composite function is "x" this means that the functions are inverses of each other.