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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 61 (page 269)

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{2}{3 + x}$

Short Answer

Expert verified

The inverse off is $\frac{2 - 3 x}{x} .$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \frac{2}{3 + x}$

We have to find its inverse and check the answer.

02

Step 2. Replace f(x) with y and interchange the variables x and y

So, $y = \frac{2}{3 + x}$

Interchange the variables

$x = \frac{2}{3 + y}$

03

Step 3. Solve for y

$x = \frac{2}{3 + y} x (3 + y) = 2 3 x + x y = 2 x y = 2 - 3 x y = \frac{2 - 3 x}{x}$

The inverse is $f^{- 1} (x) = \frac{2 - 3 x}{x}$

04

Step 4. Check the answer

We check the answer by $f^{- 1} (f (x)) a n d f (f^{- 1} (x)) .$

So,

$f^{- 1} (f (x)) = f^{- 1} (\frac{2}{3 + x}) = \frac{2 - 3 (\frac{2}{3 + x})}{(\frac{2}{3 + x})} = \frac{6 + 2 x - 6}{2} = x$

Thus, $f^{- 1} (f (x)) = x$

Now,

$f (f^{- 1} (x)) = f (\frac{2 - 3 x}{x}) = \frac{2}{3 + (\frac{2 - 3 x}{x})} = \frac{2}{\frac{3 x + 2 - 3 x}{x}} = x$

Thus, $f (f^{- 1} (x)) = x$

Therefore, our answer is right.

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Most popular questions from this chapter

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

$f (x) = \frac{2 x + 3}{x + 2}$

If $f (x) = a x + b$ and $g (x) = c x + d$ , then find: -

(a) $f 鈭� g$

(b) $g 鈭� f$

(c)the domain of $f 鈭� g$ and $g 鈭� f$

(d)the conditions for which $f 鈭� g = g 鈭� f$

True or False $f (g (x)) = f (x) . g (x)$ .

solve equation. Verify your results using a graphing utility.

$8^{- x + 14} = 16^{x}$

Evaluate each expression using the graphs of $y = f (x)$ and $y = g (x)$ shown in the figure.

(a) $(g 鈭� f) (- 1)$ (b) $(g 鈭� f) (0)$ (c) $(f 鈭� g) (- 1)$ (d) $(f 鈭� g) (4)$ .

See all solutions

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