Chapter 5: Q 41. (page 269)

Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Expert verified

Functions f and g are inverses of one another .Exclude $x = - 4$ from domain of f and $x = 2$ from domain of g.

Step by step solution

Step 1. Given information.

Given functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ .

Step 2. Verify that the functions f and g and are inverses of each other.

Note that functions f and g are inverses of each other if $f (g (x)) = x$ and $g (f (x)) = x$ .

Prove $f (g (x)) = x$ as follows.

$\begin{array}{rcl} f (g (x)) & = & f (\frac{4 x - 3}{2 - x}) \\ = & \frac{2 (\frac{4 x - 3}{2 - x}) + 3}{(\frac{4 x - 3}{2 - x}) + 4} \\ = & \frac{8 x - 6 + 6 - 3 x}{4 x - 3 + 8 - 4 x} \\ = & \frac{5 x}{5} \\ = & x \end{array}$

Prove $g (f (x)) = x$ as follows.

$\begin{array}{rcl} g (f (x)) & = & g (\frac{2 x + 3}{x + 4}) \\ = & \frac{4 (\frac{2 x + 3}{x + 4}) - 3}{2 - (\frac{2 x + 3}{x + 4})} \\ = & \frac{8 x + 12 - 3 x - 12}{2 x + 8 - 2 x - 3} \\ = & \frac{5 x}{5} \\ = & x \end{array}$

Therefore, f and g are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

Note that domain of f is $x \neq - 4$ and that of gis $x \neq 2$ .

It follows that f exists for all real numbers except for $- 4$ and g exists for all real numbers except for 2.

Therefore, $x = - 4$ and $x = 2$ need to be excluded from domain of fand g respectively.

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Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verify that the functions f and g and are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

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Verify that the functions fx=2x+3x+4and gx=4x-32-xare inverses of each other by showing that fgx=x and gfx=x. Give any values ofx that need to be excluded from the domain of f and the domain ofg.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verify that the functions f and g and are inverses of each other.

Step 3. Find values of x that need to be excluded from domain of f and g.

One App. One Place for Learning.

Most popular questions from this chapter

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Verify that the functions $f (x) = \frac{2 x + 3}{x + 4}$ and $g (x) = \frac{4 x - 3}{2 - x}$ are inverses of each other by showing that $f (g (x)) = x$ and $g (f (x)) = x$ . Give any values ofx that need to be excluded from the domain of f and the domain ofg.