Problem 14 For each pair of functions, find... [FREE SOLUTION]

91影视

Chapter 12: Problem 14

For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$. $$f(x)=1 / x^{2} ; g(x)=x+2$$

Short Answer

Expert verified

(a) $\frac{1}{9}$ (b) 3 (c) $\frac{1}{(x + 2)^2}$ (d) $\frac{1}{x^2} + 2$

Step by step solution

Identify the functions

Given two functions: $f(x) = \frac{1}{x^2}$ and $g(x) = x + 2$.

Compute $f \circ g$(1)

First find $g(1)$: \[g(1) = 1 + 2 = 3\]Next, substitute 3 into $f(x)$: \[f(g(1)) = f(3) = \frac{1}{3^2} = \frac{1}{9}\]Therefore, $(f \circ g)(1) = \frac{1}{9}$.

Compute $g \circ f$(1)

First find $f(1)$: \[f(1) = \frac{1}{1^2} = 1\]Next, substitute 1 into $g(x)$: \[g(f(1)) = g(1) = 1 + 2 = 3\]Therefore, $(g \circ f)(1) = 3$.

Compute $f \circ g$(x)

Find $g(x)$: \[g(x) = x + 2\]Then compose $f(g(x))$: \[f(g(x)) = f(x + 2) = \frac{1}{(x + 2)^2}\]Thus, $(f \circ g)(x) = \frac{1}{(x + 2)^2}$.

Compute $g \circ f$(x)

Find $f(x)$: \[f(x) = \frac{1}{x^2}\]Then compose $g(f(x))$: \[g(f(x)) = g\left( \frac{1}{x^2} \right) = \frac{1}{x^2} + 2\]Thus, $(g \circ f)(x) = \frac{1}{x^2} + 2$.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

function composition

Function composition involves creating a new function by applying one function to the result of another function. If you have two functions, say $f$ and $g$, the composition of $f$ and $g$, denoted as $f \, \circ \, g$, is defined as $f(g(x))$. This means you first apply $g$ to $x$, and then apply $f$ to the result of $g(x)$.

For example, if $f(x) = \frac{1}{x^2}$ and $g(x) = x + 2$, the composition $f \circ g$ means you first find $g(x)$, and then use this result as the input for $f$. This concept is easier to grasp with practice, starting from simple functions before progressing to more complex ones.

evaluating functions

Evaluating functions means finding the value of a function at a specific input. This process is crucial in function composition. For instance, if you need to find $(f \circ g)(1)$, you first evaluate $g$ at 1, and then use this result to evaluate $f$.

Let's break it down:

For $g(1)$, substitute 1 into $g(x)$: $g(1) = 1 + 2 = 3$.
Next, use this result to find $f(3)$: $f(3) = \frac{1}{3^2} = \frac{1}{9}$.

Therefore, $(f \, \circ \, g)(1) = \frac{1}{9}$. By following such steps attentively, you can accurately evaluate composite functions.

algebraic expressions

An algebraic expression involves numbers, variables, and operations like addition or multiplication. When dealing with composite functions, manipulating these expressions accurately is key.

For example, with our functions $f(x) = \frac{1}{x^2}$ and $g(x) = x + 2$, we crafted the composite function $(f \circ g)(x) = \frac{1}{(x + 2)^2}$. Understanding how to substitute and simplify expressions is essential for working through composite functions.

First, find $g(x)$: $g(x) = x + 2$.
Then, substitute this into $f$: $f(g(x)) = f(x + 2) = \frac{1}{(x + 2)^2}$.

Mastering these steps helps ensure the correctness of your computations.

intermediate algebra

Intermediate algebra bridges basic algebra concepts with more advanced topics. It includes working with composite functions, translating word problems into algebraic expressions, and simplifying complex expressions.

When tackling composition problems, such as finding $g \circ f$, understanding substitutions and transformations is core. For $f(x) = \frac{1}{x^2}$ and $g(x) = x + 2$, to find $(g \circ f)(x)$:

First, evaluate $f(x)$: $f(x) = \frac{1}{x^2}$.
Next, substitute this into $g$: $g(f(x)) = g\left(\frac{1}{x^2}\right) = \frac{1}{x^2} + 2$.

Thus, $(g \circ f)(x) = \frac{1}{x^2} + 2$. By mastering intermediate algebra, you ensure smooth and accurate handling of function compositions and related algebraic manipulations.

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

For each pair of functions, find (a) \((f \circ g)(1)\) (b) \((g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;\) and \((\mathbf{d})(g \circ f)(x)\). $$f(x)=1 / x^{2} ; g(x)=x+2$$

Short Answer

Step by step solution

Identify the functions

Compute \(f \circ g\)(1)

Compute \(g \circ f\)(1)

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

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

Key Concepts

function composition

evaluating functions

algebraic expressions

intermediate algebra

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Statistics

Calculus

Discrete Mathematics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.