Chapter 2: Q. 2 TB (page 208)

For each function k that follows, find functions f , g, and h so that k = f ◦ g ◦ h. There may be more than one possible answer.
$k (x) = {(x^{2} + 1)}^{3} k (x) = \sqrt{1 + {(x - 2)}^{2}} k (x) = \sqrt{\frac{1}{3 x + 1}} k (x) = \frac{1}{\sqrt{3 x + 1}}$

Short Answer

Expert verified

If $k (x) = {(x^{2} + 1)}^{3}$ and $f (x) = x^{3}, g (x) = (x + 1), & h (x) = x^{2}$ then $(f \circ g \circ h) (x) = k (x) .$

If $k (x) = \sqrt{1 + {(x - 2)}^{2}}$ and $f (x) = \sqrt{x}, g (x) = (1 + x^{2}), & h (x) = x - 2$ then $(f \circ g \circ h) (x) = k (x) .$

If $k (x) = \sqrt{\frac{1}{3 x + 1}}$ and $f (x) = \sqrt{x}, g (x) = \frac{1}{x + 1}, & h (x) = 3 x$ then $(f \circ g \circ h) (x) = k (x) .$

If $k (x) = \frac{1}{\sqrt{3 x + 1}}$ and $f (x) = \frac{1}{x}, g (x) = \sqrt{x + 1}, & h (x) = 3 x$ then $(f \circ g \circ h) (x) = k (x) .$

Step by step solution

Step 1. Given information.

Given functions are the following.

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

Step 2. Composition of the first function.

Consider $f (x) = x^{3}, g (x) = (x + 1), & h (x) = x^{2} .$

Composition of functions.

$f \circ g \circ h = f (g (h (x))) f \circ g \circ h = = f (g (x^{2})) f \circ g \circ h = = f (x^{2} + 1) f \circ g \circ h = {(x^{2} + 1)}^{3} (f \circ g \circ h) (x) = k (x)$

Step 3. Composition of the second function.

Consider $f (x) = \sqrt{x}, g (x) = (1 + x^{2}), & h (x) = x - 2 .$

Composition of functions.

role="math" localid="1648711630588" $(f \circ g \circ h) (x) = f (g (h (x))) (f \circ g \circ h) (x) = f (g (x - 2)) (f \circ g \circ h) (x) = f (1 + {(x - 2)}^{2}) (f \circ g \circ h) (x) = \sqrt{1 + {(x - 2)}^{2}} (f \circ g \circ h) (x) = k (x)$

Step 4. Composition of the third function.

Consider $f (x) = \sqrt{x}, g (x) = \frac{1}{x + 1}, & h (x) = 3 x .$

Composition of functions.

$(f \circ g \circ h) (x) = f (g (h (x))) (f \circ g \circ h) (x) = f (g (3 x)) (f \circ g \circ h) (x) = f (\frac{1}{3 x + 1}) (f \circ g \circ h) (x) = \sqrt{\frac{1}{3 x + 1}} (f \circ g \circ h) (x) = k (x)$

Step 5. Composition of the fourth function.

Consider $f (x) = \frac{1}{x}, g (x) = \sqrt{x + 1}, & h (x) = 3 x .$

Composition of functions.

$(f \circ g \circ h) (x) = f (g (h (x))) (f \circ g \circ h) (x) = f (g (3 x)) (f \circ g \circ h) (x) = f (\sqrt{3 x + 1}) (f \circ g \circ h) (x) = \frac{1}{\sqrt{3 x + 1}} (f \circ g \circ h) (x) = k (x) .$

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For each function k that follows, find functions f , g, and h so that k = f ◦ g ◦ h. There may be more than one possible answer.
$k (x) = {(x^{2} + 1)}^{3} k (x) = \sqrt{1 + {(x - 2)}^{2}} k (x) = \sqrt{\frac{1}{3 x + 1}} k (x) = \frac{1}{\sqrt{3 x + 1}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Composition of the first function.

Step 3. Composition of the second function.

Step 4. Composition of the third function.

Step 5. Composition of the fourth function.

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For each function k that follows, find functions f , g, and h so that k = f ◦ g ◦ h. There may be more than one possible answer.k(x)=x2+13k(x)=1+x-22k(x)=13x+1k(x)=13x+1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Composition of the first function.

Step 3. Composition of the second function.

Step 4. Composition of the third function.

Step 5. Composition of the fourth function.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

For each function k that follows, find functions f , g, and h so that k = f ◦ g ◦ h. There may be more than one possible answer.
$k (x) = {(x^{2} + 1)}^{3} k (x) = \sqrt{1 + {(x - 2)}^{2}} k (x) = \sqrt{\frac{1}{3 x + 1}} k (x) = \frac{1}{\sqrt{3 x + 1}}$