Chapter 5: Q. 83 (page 270)

Find the inverse of the linear function $f (x) = m x + b m \neq 0$

Short Answer

Expert verified

The inverse of the function $f (x) = m x + b m \neq 0$ is $f^{- 1} (x) = \frac{x - b}{m} m \neq 0$

Step by step solution

Step 1. Given data

The given function is

$f (x) = m x + b m \neq 0$

Step 2. interchanging variables

Replace $f (x)$ with y and interchange x and y

role="math" localid="1646177714686" $f (x) = m x + b y = m x + b x = m y + b$

Step 3. The inverse of the function

Solve the equation for y

$x = m y + b x - b = m y \frac{x - b}{m} = y y = \frac{x - b}{m}$

replace ywith $f^{- 1} (x)$

role="math" localid="1646177950266" $f^{- 1} (x) = \frac{x - b}{m} m \neq 0$

Step 4. Verification

Determine $f (f^{- 1} (x))$

$f (f^{- 1} (x)) = f (\frac{x - b}{m}) f (f^{- 1} (x)) = m (\frac{x - b}{m}) + b f (f^{- 1} (x)) = x - b + b f (f^{- 1} (x)) = x$

Determine $f^{- 1} (f (x))$

$f^{- 1} (f (x)) = f^{- 1} (m x + b) f^{- 1} (f (x)) = \frac{(m x + b) - b}{m} f^{- 1} (f (x)) = \frac{m x}{m} f^{- 1} (f (x)) = x$

$f^{- 1} (f (x)) = x & f (f^{- 1} (x)) = x$ so inverse function is correct

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Find the inverse of the linear function $f (x) = m x + b m \neq 0$

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

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Find the inverse of the linear functionf(x)=mx+bm≠0

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Find the inverse of the linear function $f (x) = m x + b m \neq 0$