/*! 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 15 Evaluate each expression using t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

Evaluate each expression using the given table of values:$$\begin{array}{|c|c|c|c|c|c|}\hline x & -2 & -1 & 0 & 1 & 2 \\\\\hline f(x) & 1 & 0 & -2 & 1 & 2 \\\\\hline g(x) & 2 & 1 & 0 & -1 & 0 \\\\\hline\end{array}.$$ a. \(f(g(-1))\) b. \(g(f(0))\) c. \(f(f(-1))\) d. \(g(g(2))\) e. \(g(f(-2))\) f. \(f(g(1))\)

Short Answer

Expert verified
a. 1, b. 2, c. -2, d. 0, e. -1, f. 0

Step by step solution

01

Evaluate f(g(-1))

1. First, find the value of \( g(-1) \). Using the table, we have \( g(-1) = 1 \). 2. Next, evaluate \( f(g(-1)) \) by finding \( f(1) \) in the table. We get \( f(1) = 1 \). 3. Therefore, \( f(g(-1)) = 1 \).
02

Evaluate g(f(0))

1. First, find \( f(0) \). From the table, \( f(0) = -2 \). 2. Next, evaluate \( g(f(0)) \) by finding \( g(-2) \) in the table. We have \( g(-2) = 2 \). 3. Therefore, \( g(f(0)) = 2 \).
03

Evaluate f(f(-1))

1. First, find \( f(-1) \). According to the table, \( f(-1) = 0 \). 2. Next, evaluate \( f(f(-1)) \) by finding \( f(0) \). We find \( f(0) = -2 \) in the table. 3. Therefore, \( f(f(-1)) = -2 \).
04

Evaluate g(g(2))

1. First, find \( g(2) \). From the table, \( g(2) = 0 \). 2. Next, evaluate \( g(g(2)) \) by finding \( g(0) \) in the table. We find \( g(0) = 0 \). 3. Therefore, \( g(g(2)) = 0 \).
05

Evaluate g(f(-2))

1. First, find \( f(-2) \). According to the table, \( f(-2) = 1 \). 2. Next, evaluate \( g(f(-2)) \) by finding \( g(1) \). We get \( g(1) = -1 \). 3. Therefore, \( g(f(-2)) = -1 \).
06

Evaluate f(g(1))

1. First, find \( g(1) \). According to the table, \( g(1) = -1 \). 2. Next, evaluate \( f(g(1)) \) by finding \( f(-1) \) in the table. We have \( f(-1) = 0 \). 3. Therefore, \( f(g(1)) = 0 \).

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 Evaluation
Function evaluation is the process of determining the output value of a function for a given input. In simple terms, it involves substituting the input value into the function's formula to find the result. For example, if we have a function \( f(x) = 2x + 3 \), to evaluate \( f \) at \( x = 2 \), we substitute 2 into the expression, resulting in \( f(2) = 2(2) + 3 = 4 + 3 = 7 \).

The main steps include:
  • Identify the input value.
  • Substitute the input into the function.
  • Perform the arithmetic operations to find the output.
Function evaluation is fundamental as it helps us understand how the function behaves for different inputs, which is crucial for problem-solving in mathematics.
Function Table
A function table is a powerful tool to represent the values of a function for specific inputs. It is a structured way to display how the function maps each input to its corresponding output. These tables usually have two rows or columns: one for input values (\( x \)) and one for output values (\( f(x) \) or \( g(x) \)).

With a function table:
  • Each row represents a unique input-output pair.
  • The table clearly shows how changing the input affects the output.
This makes it easier to evaluate functions at specific points and is particularly useful when dealing with composite functions.
Function Composition
Function composition involves combining two functions to create a new function. This means applying one function to the results of another. It's written as \( (f \circ g)(x) \), which reads "\( f \) composed with \( g \) of \( x \)." This process is like feeding the output of \( g(x) \) as the input to \( f(x) \).

For instance, if \( f(x) = x + 2 \) and \( g(x) = 3x \), then \( f(g(x)) \) would mean substituting \( 3x \) into \( f(x) \), making it \( f(g(x)) = 3x + 2 \). Here are steps to compose functions:
  • Determine \( g(x) \) from the input value.
  • Use this output as the input for \( f \).
  • Evaluate \( f(g(x)) \) to find the final result.
Function composition is an important concept as it extends the idea of function evaluation, allowing more complex operations between functions.
Table of Values
A table of values is essentially a dataset used to represent the function's inputs and their corresponding outputs. It's crucial in understanding the behavior of a function. Each entry (or "cell") in the table indicates a relationship between the \( x \) values and the function outputs, like \( f(x) \) or \( g(x) \), providing a snapshot of how the function operates at certain key points.

How it helps:
  • Visualizes numerous function evaluations at once.
  • Assists in identifying patterns, such as linearity or periodicity.
  • Useful for assessing how changes in one function's output influence another when doing compositions.
In more complex exercises, you'll often refer to a table of values to quickly find outputs without recalculating functions for every single input, making it an essential aid in mathematics education.

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 the functions. $$y=\frac{1}{x-2}.$$

You will explore graphically the general sine function $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$ as you change the values of the constants \(A, B, C,\) and \(D .\) Use a CAS or computer grapher to perform the steps in the exercises. Set the constants \(A=3, B=6, D=0\). a. Plot \(f(x)\) for the values \(C=0,1,\) and 2 over the interval \(-4 \pi \leq x \leq 4 \pi .\) Describe what happens to the graph of the general sine function as \(C\) increases through positive values. b. What happens to the graph for negative values of \(C ?\) c. What smallest positive value should be assigned to \(C\) so the graph exhibits no horizontal shift? Confirm your answer with a plot.

Graph the functions. $$y=\frac{1}{x+2}.$$

The table shows the average residential and transportation prices for energy consumption in the United States for the years \(2000-2008,\) as reported by the U.S. Department of Energy. The prices are given as dollars paid for one million BTU (British thermal units) of consumption. $$\begin{array}{lcc}\hline \text { Year } & \text { Residential (\$) } & \text { Transportation (\$) } \\\\\hline 2000 & 15 & 10 \\\2001 & 16 & 10 \\\2002 & 15 & 9 \\\2003 & 16 & 11 \\\2004 & 18 & 13 \\\2005 & 19 & 16 \\\2006 & 21 & 19 \\\2007 & 21 & 20 \\\2008 & 23 & 25 \\\\\hline\end{array}$$ a. Make a scatterplot of the data sets. b. Find and plot a regression line for each set of data points, and superimpose the lines on their scatterplots. c. What do you estimate as the average energy price for residential and transportation use for a million BTU in year \(2017 ?\) d. In looking at the trend lines, what do you conclude about the rising costs of energy across the two sectors of usage?

Find the function values. $$\cos ^{2} \frac{5 \pi}{12}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

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.