/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find \((f \circ g)(x)\) and \((g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 23

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$f(x)=\sqrt{x}, g(x)=-5 x+2$$

Short Answer

Expert verified
\((f \circ g)(x) = \sqrt{-5x + 2}\); \((g \circ f)(x) = -5\sqrt{x} + 2\).

Step by step solution

01

Identify Each Function

We first recognize the functions provided to us: \( f(x) = \sqrt{x} \) and \( g(x) = -5x + 2 \). Our task is to find two composite functions: \((f \circ g)(x)\) and \((g \circ f)(x)\).
02

Compose by Substitution: First Function

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). This means we must evaluate \(f(g(x))\). Given \(g(x) = -5x + 2\), substitute this into \(f(x) = \sqrt{x}\) to get \(f(g(x)) = \sqrt{-5x + 2}\). So, \((f \circ g)(x) = \sqrt{-5x + 2}\).
03

Compose by Substitution: Second Function

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). This means replacing \(x\) in \(g(x) = -5x + 2\) with \(f(x) = \sqrt{x}\). So, we have \(g(f(x)) = -5(\sqrt{x}) + 2\). Thus, \((g \circ f)(x) = -5\sqrt{x} + 2\).

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.

Understanding the Square Root Function
The square root function is a fundamental mathematical function. It's defined as \( f(x) = \sqrt{x} \). This function takes any non-negative number \( x \) and transforms it into its square root. The graph of the square root function is a gentle curve that begins at the origin \( (0,0) \) and rises steadily but slowly to the right.

  • Domain: The set of all non-negative real numbers, since we cannot take the square root of a negative number in real numbers.
  • Range: The set of all non-negative real numbers as well.
  • Behavior: As \( x \) increases, \( \sqrt{x} \) increases, but at a decreasing rate.
Understanding this behavior is crucial when it comes to composing functions, as it will influence the output based on the input values given by another function.
Grasping the Linear Function
A linear function is one of the simplest types of functions. It has the form \( g(x) = ax + b \), where \( a \) and \( b \) are constants. In the case of \( g(x) = -5x + 2 \), \( a = -5 \) and \( b = 2 \). Linear functions produce a straight line when graphed. They have a constant rate of change, represented by the slope.

  • Slope: The slope \( a \) (here, \(-5\)) tells us that for every 1 unit increase in \( x \), \( g(x) \) decreases by 5 units.
  • Y-intercept: This is the point where the line crosses the y-axis. For \( g(x) = -5x + 2 \), the line intercepts at \( y = 2 \).
The simplicity of the linear function makes it a common choice in function composition, providing straightforward inputs which can transform more complex functions like the square root function.
Unpacking Composite Functions
Composite functions are functions that are formed by using one function as the input for another. Notationally, the composition of two functions \( f \) and \( g \) is expressed as \( (f \circ g)(x) \), which means \( f(g(x)) \). This involves replacing \( x \) in the first function with the whole of the second function.

  • Order of Composition: The order is crucial. \((f \circ g)(x)\) is different from \((g \circ f)(x)\).
  • Input Transformation: The result of the inner function \( g(x) \) becomes the input for \( f(x) \) in \((f \circ g)(x)\).
Taking an example from our exercise, composing \( f(x) \) and \( g(x) \):
  • \((f \circ g)(x) = \sqrt{-5x + 2}\) involves substituting \( g(x) = -5x + 2 \) into the square root function \( f(x) = \sqrt{x} \).
  • \((g \circ f)(x) = -5\sqrt{x} + 2\) means substituting \( f(x) = \sqrt{x} \) into the linear function \( g(x) = -5x + 2 \).
Mastering composite functions opens up a new realm of calculating complex equations and is incredibly powerful when analyzing mathematical relationships.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log (x+2) $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x}+2 $$

Explain why the graph of the function \(y=\log _{b} x\) contains the point \((1,0)\) no matter what \(b\) is.

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=e^{x}-3 $$

Find the value of each logarithmic expression. See Examples 3 and 5. $$ \log _{9} 9 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.