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Chapter 9: Problem 58

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(4) $$

Short Answer

Expert verified
(g \circ f)(4) = 110.

Step by step solution

01

Understand the composition of functions

The notation \((g \circ f)(4)\) means we need to find \(g(f(4))\). This involves first evaluating the inner function \(f(x)\) at \(x = 4\) and then using that result to evaluate \(g(x)\).
02

Evaluate \(f(4)\)

Substitute \(x = 4\) into the expression for \(f(x)\). We have: \(f(x) = 3x - 2\). Therefore, \(f(4) = 3(4) - 2 = 12 - 2 = 10\).
03

Evaluate \(g(10)\)

Now that we know \(f(4) = 10\), substitute \(x = 10\) into the expression for \(g(x)\), which is \(g(x) = x^2 + x\). Therefore, \(g(10) = 10^2 + 10 = 100 + 10 = 110\).
04

Write the final answer

Since \(g(f(4)) = g(10) = 110\), we conclude that \((g \circ f)(4) = 110\).

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Key Concepts

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

Evaluating Functions
Evaluating functions means finding the output of a function for a given input. To evaluate a function, you substitute the input value into the function鈥檚 equation. Let's consider a function like \(f(x) = 3x - 2\). If you need to evaluate \(f\) at \(x = 4\), replace \(x\) with \(4\) in the equation. Here鈥檚 how it works:
  • Start with the equation: \(f(x) = 3x - 2\).
  • Substitute \(x = 4\): \(f(4) = 3(4) - 2\).
  • Calculate the result: \(12 - 2 = 10\).
So, \(f(4) = 10\). This process is simple and helps determine how a function changes for specific inputs. Evaluating is a crucial first step in more complex operations like composition.
Polynomial Functions
Polynomial functions are expressions involving sums of powers of variables multiplied by coefficients. They appear often in mathematics and have many different forms. For instance, \(g(x) = x^2 + x\) is a polynomial function. Let鈥檚 break down this example:
  • The highest power of \(x\) is \(2\), which makes it a quadratic function.
  • The terms are \(x^2\) and \(x\) with coefficients of \(1\) for both.
  • It is expressed in standard form as \(ax^2 + bx + c\), where \(a = 1\), \(b = 1\), and \(c = 0\).
Polynomial functions can be evaluated just like simpler functions. Substituting a specific \(x\) value lets you find the output. For example, if you substitute \(x = 10\) in \(g(x) = x^2 + x\), you'd get \(g(10) = 10^2 + 10 = 110\). This evaluates the function at that point, yielding a specific result.
Composition of Functions
The composition of functions is a method for combining two functions. Here, one function鈥檚 output becomes the input for another. This is commonly represented as \((g \circ f)(x)\), meaning apply \(f(x)\) first, then \(g\). It鈥檚 like using a tool within another tool. For example, with \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\):
  • First, evaluate \(f(x)\) at a point. Suppose \(x = 4\): we have \(f(4) = 10\).
  • Then use this output as the input for \(g(x)\). So here \(g(10) = 10^2 + 10 = 110\).
  • The final result for \((g \circ f)(4)\) is \(g(f(4)) = 110\).
This method lets you blend functions to create new outputs. It鈥檚 useful for complex models and analysis, simplifying multiple operations into one seamless expression.

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Most popular questions from this chapter

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. See Example 5. $$ h(x)=\frac{1}{3 x-16} $$

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(x) $$

Assume that there are no deposits or withdrawals. Comparison of Compounding Methods. An initial deposit of \(\$ 5,000\) grows at an annual rate of \(8.5 \%\) for 5 years. Compare the final balances resulting from annual compounding and continuous compounding.

Evaluate each expression without using a calculator. $$ \ln \frac{1}{e^{3}} $$

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$ \ln 0.675 $$
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