Chapter 5.2, Problem 52AYU

a.

To determine

To find: The function f is one-to-one.

a. Find its inverse function f- 1 and check your answer.

Answer to Problem 52AYU

Solution:

a. $f^{-1}\left(x\right)=\frac{\left(1-x\right)}{3}$

Explanation of Solution

i.e, A function f is one-to-one if for all a, b $\in A$ and a $\ne$ b $\Rightarrow$ f(a ) $\ne$ f(b) $\Rightarrow$ x $\ne$ y

Now, $f\left(a\right)=x$ . $\Rightarrow f^{-1}\left(f\left(a\right)\right)=f^{-1}\left(x\right)$

  $\Rightarrow a=f^{-1}\left(x\right)$ .

Given:

f ( x ) = 1 - 3x

Calculation:

a.Let $f\left(x\right)=1-3x=y$ .

• i.e., $1-3x=y$
• $3x=1-y$
• $x=\frac{1-y}{3}$
• Change x to y and y to x .
• $y=\frac{1-x}{3}$
• Rewriting this, we get $f^{-1}\left(x\right)=\frac{1-x}{3}$ .

  $\left(f\cdot f^{-1}\right)\left(x\right)=f(f^{-1}\left(x\right))$

• $\left(f\cdot f^{-1}\right)\left(x\right)=f\left(\frac{1-x}{3}\right)$
• $\left(f\cdot f^{-1}\right)\left(x\right)=1-3\left(\frac{1-x}{3}\right)$
• $\left(f\cdot f^{-1}\right)\left(x\right)=1-1+x$
• $\left(f\cdot f^{-1}\right)\left(x\right)=x$

Also,

• $\left(f^{-1}\cdot f\right)\left(x\right)=f^{-1}\left(f\left(x\right)\right)$
• $\left(f^{-1}\cdot f\right)\left(x\right)=f^{-1}\left(1-3x\right)$
• $\left(f^{-1}\cdot f\right)\left(x\right)=\frac{1-\left(1-3x\right)}{3}$
• $\left(f^{-1}\cdot f\right)\left(x\right)=\frac{1-1+3x}{3}$
• $\left(f^{-1}\cdot f\right)\left(x\right)=\frac{3x}{3}$
• $\left(f^{-1}\cdot f\right)\left(x\right)=x$
• Here we verified that $\left(f\cdot f^{-1}\right)\left(x\right)=x=\left(f^{-1}\cdot f\right)\left(x\right)$ .

b.

To determine

To find: The function f is one-to-one.

b. Find the domain and the range of f and f- 1 .

Answer to Problem 52AYU

Solution:

b. The domain of f = Range of f −1 = all real numbers of x .

The range of f = Domain of f −1 = all real numbers of x .

Explanation of Solution

Given:

f ( x ) = 1 - 3x

Calculation:

b. The domain of f = Range of f −1 = all real numbers of x .

The range of f = Domain of f −1 = all real numbers of x .

To find: The function f is one-to-one.

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