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Chapter 5: Problem 37

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5\). \((g \circ f)(6)\)

Short Answer

Expert verified
(g \circ f)(6) = 83

Step by step solution

01

Understand Composite Functions

The notation \(g \circ f\) means applying function \(f\) first and then applying function \(g\) to the result of \(f\). So, \( (g \circ f)(6) = g(f(6))\).
02

Calculate \(f(6)\)

Substitute \ x = 6 \ into the function \( f(x) \). \( f(x) = x^2 + 4 \). So, \( f(6) = 6^2 + 4 = 36 + 4 = 40 \).
03

Calculate \(g(40)\)

Now, substitute the value \( 40 \) into the function \( g(x) \). \( g(x) = 2x + 3 \). So, \( g(40) = 2(40) + 3 = 80 + 3 = 83 \).

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Key Concepts

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

function notation
Functions are mathematical expressions that describe a relationship between two variables, typically represented as \( f(x) \). This notation means that you have a function called \( f \) which depends on \( x \), the variable.
For example, in the exercise given, we have three functions: \( f(x) = x^2 + 4 \), \( g(x) = 2x + 3 \), and \( h(x) = x - 5 \).
These equations show exactly what calculation to perform on the variable \( x \). When you see these functions, you substitute the value of \( x \) into the expression to find the result. Here鈥檚 a quick breakdown:
  • \( f(x) = x^2 + 4 \) means you square \( x \) and then add 4.
  • \( g(x) = 2x + 3 \) means you multiply \( x \) by 2 and then add 3.
  • \( h(x) = x - 5 \) means you subtract 5 from \( x \).
This notation is essential when working with composite functions, which we'll delve into next.
function composition
Function composition combines two functions into one by inputting the result of the first function into the second. The notation \( (g \circ f)(x) \) indicates that you first apply function \( f \) to \( x \) and then apply function \( g \) to the result of \( f(x) \).
In our exercise, we need to find \( (g \circ f)(6) \), which means applying function \( f \) to 6, and then plugging the result into function \( g \).
Let's break this down:
  • First, we find \( f(6) \). Since \( f(x) = x^2 + 4 \), we substitute 6 for \( x \) to get \( f(6) = 6^2 + 4 = 40 \).
  • Next, we use this result as an input for \( g \). Substituting 40 into \( g(x) = 2x + 3 \), we find that \( g(40) = 2 \times 40 + 3 = 83 \).
Therefore, \( (g \circ f)(6) = 83 \). Composing functions is a fundamental concept in calculus and algebra, allowing us to build more complex functions from simpler ones.
evaluation of functions
Evaluating a function involves substituting a specific value for the variable \( x \) and calculating the result. This process provides the output of the function for that particular input.
Let's revisit the functions in the exercise:
  • For \( f(x) = x^2 + 4 \), evaluating at \( x = 6 \) means computing \( f(6) \). So, \( f(6) = 6^2 + 4 = 36 + 4 = 40 \).
  • For \( g(x) = 2x + 3 \), when we evaluate it at \( x = 40 \), we compute \( g(40) = 2 \times 40 + 3 = 80 + 3 = 83 \).
The result of each function tells you the output corresponding to the input you provided.
This is especially useful in solving real-world problems where functions are used to model relationships between variables. By mastering the evaluation of functions, you can better understand and interpret these relationships.

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Most popular questions from this chapter

Write each number in scientific notation. $$ 1600 $$

Solve each problem. In the early years of the Powerball Lottery, a player would choose five numbers from 1 through 49 and one number from 1 through \(42 .\) It can be shown that there are about \(8.009 \times 10^{7}\) different ways to do this. Suppose that a group of 2000 persons decided to purchase tickets for all these numbers and each ticket cost \(\$ 1.00 .\) How much should each person have expected to pay?

Solve each problem. In January 2010 , the population of the United States was approximately 308.4 million. (a) Write the January 2010 population using scientific notation. (b) Write \(S 1\) trillion, that is, \(\$ 1,000,000,000,000,\) using scientific notation. (c) Using your answers from parts (a) and (b), calculate how much each person in the United States in the year 2010 would have had to contribute in order to make someone a trillionaire. Write this amount in standard notation to the nearest dollar.

Solve each problem. In \(2009,\) the population of Costa Rica was approximately \(4.25 \times 10^{6} .\) The population density was 83.3 people per square kilometer. (a) Write the population density in scientific notation. (b) To the nearest square kilometer, what is the area of Costa Rica?

Solve each problem. In \(2009,\) the national debt of the U.S. government was about \(\$ 11.5\) trillion. Using 307.0 million as the population for that year, about how much was this per American? Write the amount in standard notation to the nearest dollar.
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