30.3k views
5 votes
Verify the equation is correct and fill in the blanks for B

Verify the equation is correct and fill in the blanks for B-example-1
User BlaShadow
by
8.2k points

1 Answer

3 votes

Given that the function


f(x)\text{ =13x}

is a one-to-one function,

A) Equation for f⁻¹(x), the inverse function.

From the function f(x), interchange x and f(x).

thus,


\begin{gathered} f(x)\text{ = 13x} \\ in\text{terchanging x and f(x), we have} \\ x\text{ = 13f(x)} \\ \text{divide both sides by 13} \\ f(x)\text{ = }(x)/(13) \end{gathered}

thus, equation for f⁻¹(x), the inverse function is


f(x)\text{ = }(x)/(13)

B) To show that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Starting with f(f⁻¹(x)):

Evaluating f(f⁻¹(x)) involves substituting the f⁻¹(x) function into the f(x) function.

In this case, let f⁻¹(x) be z(x).

Thus, we have f(f⁻¹(x)) to be f(z).


\begin{gathered} z(x)=f^(-1)(x)\text{ = }(x)/(13) \\ f(x)\text{ = }13x \\ \text{thus, substituting the z(x) function into the f(x) function, we have} \\ f(z(x))\text{ = }13((x)/(13)) \\ \Rightarrow x \\ \end{gathered}

For f⁻¹(f(x)):

Similarly, evaluating f⁻¹(f(x)) involves substituting the f(x) function into the f⁻¹(x) function.

Thus, we have


\begin{gathered} f(x)\text{ = 13x} \\ f^(-1)(x)\text{ = }(x)/(13) \\ \text{thus, substituting f(x) into }f^(-1)(x)\text{ gives} \\ f^(-1)(f(x))\text{ = }(1)/(13)(13x) \\ \Rightarrow x \\ \end{gathered}

Hence,


\begin{gathered} f(f^(-1)(x))\text{ = }f((x)/(13)) \\ \text{ = x} \end{gathered}
\begin{gathered} f^(-1)(f(x))\text{ = }f^(-1)(13x) \\ =\text{ x} \end{gathered}

User Kyle Krzeski
by
8.5k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories