Chapter 5: Q 57. (page 269)

The function $f (x) = \frac{4}{x}$ is one-to-one. Find its inverse and check the answer. Graph $f, f^{- 1}$ and $y = x$ on the same coordinate axes.

Short Answer

Expert verified

Inverse of the function $f (x) = \frac{4}{x}$ is $f^{- 1} (x) = \frac{4}{x}$ . Graph is shown below.

Step by step solution

Step 1. Given information.

Given a one-to-one function $f (x) = \frac{4}{x}$ .

Step 2. Find the inverse of the function f and verify.

To find the explicit form of the inverse solve $x = \frac{4}{y}$ for y.

It can be seen that $y = \frac{4}{x}$ .

Note that if $f (g (x)) = x$ and $g (f (x)) = x$ , then fand g are inverse of one another.

Prove $f (f^{- 1} (x)) = x$ as follows.

$\begin{array}{rcl} f (f^{- 1} (x)) & = & f (\frac{4}{x}) \\ = & \frac{4}{\frac{4}{x}} \\ = & x \end{array}$

Prove $f^{- 1} (f (x)) = x$ as follows.

$\begin{array}{rcl} f^{- 1} (f (x)) & = & f^{- 1} (\frac{4}{x}) \\ = & \frac{4}{\frac{4}{x}} \\ = & x \end{array}$

Therefore, $f^{- 1} (x) = \frac{4}{x}$ is inverse of $f (x) = \frac{4}{x}$ .

Step 3. Graph f,f-1 and y=x on the same coordinate axes.

Graph $f^{- 1} (x) = \frac{4}{x}, f (x) = \frac{4}{x}$ and $y = x$ as follows.

Note that function f and its inverse is one and the same. So, the graph for both coincides.

Broken blue color line represents the function f and unbroken orange color line represents its inverse.

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The function $f (x) = \frac{4}{x}$ is one-to-one. Find its inverse and check the answer. Graph $f, f^{- 1}$ and $y = x$ on the same coordinate axes.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the inverse of the function f and verify.

Step 3. Graph f,f-1 and y=x on the same coordinate axes.

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The function fx=4xis one-to-one. Find its inverse and check the answer. Graph f,f-1 and y=x on the same coordinate axes.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the inverse of the function f and verify.

Step 3. Graph f,f-1 and y=x on the same coordinate axes.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Pure Maths

Geometry

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

The function $f (x) = \frac{4}{x}$ is one-to-one. Find its inverse and check the answer. Graph $f, f^{- 1}$ and $y = x$ on the same coordinate axes.