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Chapter 5: Q. 85 (page 309)

If f(x)=lnx,g(x)=ex, and h(x)=x2, find:

(a)role="math" localid="1647337775231" f∘g(x). What is the domain of f∘g?

(b) g∘f(x). What is the domain of g∘f ?

(c) (f∘g)(5)

(d)(f∘h)(x). What is the domain of f∘h?

(e) f∘he

Short Answer

Expert verified

(a) f∘g(x)=x

The domain of f∘gis -∞,∞

(b)g∘f(x)=x

The domain of g∘fis -∞,∞

(c) f∘g(5)=5

(d) f∘h(x)=lnx2

Domain of f∘his -∞,0∪0,∞

(e)f∘he=2

Step by step solution

01

Step 1. Given information

The Given functions are

f(x)=lnxg(x)=exh(x)=x2

02

Step 2. Part (a)

Composition of function f∘g(x)

f∘g(x)=f(g(x))f∘g(x)=lng(x)f∘g(x)=lnexf∘g(x)=x

So the domain off∘gis-∞,∞

03

Step 3. Part (b)

Composition of function g∘fx

g∘fx=g(f(x))g∘fx=ef(x)g∘fx=elnxg∘fx=x

So the domain ofg∘fis-∞,∞

04

Step 4. Part (c)

Substitute x=5in f∘g(x)

f∘g(x)=xf∘g(5)=5

05

Step 5. Part (d)

Composition of function f∘h(x)

f∘h(x)=f(h(x))f∘h(x)=f(x2)f∘h(x)=lnx2

So the domain off∘his-∞,0∪0,∞

06

Step 6. Part (e)

Substitute x=ein role="math" f∘h

f∘h(x)=lnx2f∘h(e)=lne2f∘h(e)=2

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Most popular questions from this chapter

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