Problem 40 Find (a) $f \circ g$ and (b) \... [FREE SOLUTION]

Chapter 1: Problem 40

Find (a) $f \circ g$ and (b) $g \circ f .$ Find the domain of each function and each composite function. $$f(x)=|x-4|, \quad g(x)=3-x$$

Short Answer

Expert verified

$f \circ g (x) = |x-1|$ , $g \circ f (x) = 3 - |x-4|$. The domain for all functions $f(x)$, $g(x)$, $f \circ g(x)$ and $g \circ f(x)$ is all real numbers.

Step by step solution

Compute $f \circ g (x)$

The composite function $f \circ g(x)$ is read as 'f of g of x', which means we replace every x in the function f with the function g. Thus, $f \circ g (x) = f (g(x)) = f (3-x)$. By substituting $3-x$ into f(x), we get $f(g(x)) = |(3-x)-4|= |3-x-4|=|x-1|$.

Compute $g \circ f (x)$

The composite function $g \circ f (x)$ is read as 'g of f of x', which means we replace every x in the function g with the function f. Thus, $g \circ f (x) = g (f(x)) = g (|x-4|)$. By substituting $|x-4|$ into g(x), we get $g(f(x)) = 3 - |x-4|$.

Find Domains

The domain of a function is the set of all possible x-values which will make the function 'work', and will output real y-values. For the individual functions, the domain of $f(x)=|x-4|$ is all real numbers as the absolute function can take all real number values making x: $-\infty < x < \infty$. Similarly, for $g(x)=3-x$, the domain is all real numbers as x can be any real number. For the composite functions, as both functions have the domain of all real numbers, the composite function $f \circ g(x)=|x-1|$ and $g \circ f (x) = 3 - |x-4|$ will also have the same domain, all real numbers.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=|x-4|, \quad g(x)=3-x$$

Short Answer

Step by step solution

Compute \(f \circ g (x)\)

Compute \(g \circ f (x)\)

Find Domains

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.