Chapter 5: Q 50. (page 257)

In Problems 45–52, show that $(f \circ g) (x) = (g \circ f) (x) = x$
$f (x) = 4 - 3 x; g (x) = \frac{1}{3} (4 - x)$

Short Answer

Expert verified

$(f \circ g) (x) = x (g \circ f) (x) = x$

Therefore, $(f \circ g) (x) = (g \circ f) (x) = x$ .

Step by step solution

Step 1. Given information.

The given composite function is:

$f (x) = 4 - 3 x g (x) = \frac{1}{3} (4 - x)$

When we are given two functions f and g, the composite function which is denoted by $f \circ g$ is defined by $(f \circ g) (x) = f (g (x))$ .

Step 2. Find (f∘g)(x).

$(f \circ g) (x) = f (g (x))$

Now substitute $g (x) = \frac{1}{3} (4 - x)$ in the function $f (g (x))$ ,

Then the function will become $f (\frac{1}{3} (4 - x))$ .

$f (\frac{1}{3} (4 - x)) = 4 - 3 (\frac{1}{3} (4 - x)) = 4 - 3 (\frac{4 - x}{3}) = 4 - 4 + x = x$

Therefore, $(f \circ g) (x) = x$ .

Step 3. Find (g∘f)(x).

$(g \circ f) (x) = g (f (x))$

Substitute $f (x) = 4 - 3 x$ in the function $g (f (x))$ ,

localid="1646305216024" $(g \circ f) (x) = g (f (x)) = g (4 - 3 x) = \frac{1}{3} (4 - (4 - 3 x)) = \frac{1}{3} (4 - 4 + 3 x) = x$

Therefore, $(g \circ f) (x) = x$ .

It is shown that $(f \circ g) (x) = (g \circ f) (x) = x$ .

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In Problems 45–52, show that $(f \circ g) (x) = (g \circ f) (x) = x$
$f (x) = 4 - 3 x; g (x) = \frac{1}{3} (4 - x)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find (f∘g)(x).

Step 3. Find (g∘f)(x).

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In Problems 45–52, show that (f∘g)(x)=(g∘f)(x)=xf(x)=4-3x;g(x)=13(4-x)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find (f∘g)(x).

Step 3. Find (g∘f)(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

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Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 45–52, show that $(f \circ g) (x) = (g \circ f) (x) = x$
$f (x) = 4 - 3 x; g (x) = \frac{1}{3} (4 - x)$