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Chapter 5: Problem 16

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ g(2) / f(2) $$

Short Answer

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Step by step solution

01

- Compute the value of g(2)

Substitute 2 into the function for g(x):g(x) = x^2 + 2g(2) = 2^2 + 2g(2) = 4 + 2g(2) = 6
02

- Compute the value of f(2)

Substitute 2 into the function for f(x):f(x) = -3x + 1f(2) = -3(2) + 1f(2) = -6 + 1f(2) = -5
03

- Divide g(2) by f(2)

Now, divide the value of g(2) by the value of f(2):g(2) / f(2) = 6 / (-5)g(2) / f(2) = -6/5

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Key Concepts

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

Function Evaluation
Function evaluation involves finding the value of a function at a specific point. For example, if we have a function \(f(x)\), we find \(f(a)\) by substituting \(a\) into the function in place of \(x\).

In our exercise, we need to evaluate \(g(2)\) and \(f(2)\). Let's break this down:
  • For \(g(x) = x^2 + 2\), we substitute 2 where \(x\) is: \[ g(2) = 2^2 + 2 = 4 + 2 = 6 \]
  • For \(f(x) = -3x + 1\), we substitute 2 where \(x\) is: \[ f(2) = -3(2) + 1 = -6 + 1 = -5 \]
By following these steps, we correctly find the values of \(f(2)\) and \(g(2)\). This is a basic yet crucial skill in functions that helps understand their behavior at specific points.
Substitution Method
The substitution method is a technique where we replace a variable in an equation or expression with a given value. This method is particularly useful when working with functions.

In the given exercise, we used substitution to evaluate the functions at \(x = 2\):
  • For \(g(x) = x^2 + 2\), substituting \(x = 2\), we performed: \[ g(2) = 2^2 + 2 \]
  • For \(f(x) = -3x + 1\), substituting \(x = 2\), we performed: \[ f(2) = -3(2) + 1 \]

Substitution allows us to simplify the function and directly compute its value at the desired point. It鈥檚 a straightforward process once you identify the value you need to substitute.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are both polynomials. In our exercise, after evaluating the functions, we need to divide \(g(2)\) by \(f(2)\).

Let's look at the final step:
  • We first found that \(g(2) = 6\) and \(f(2) = -5\).
  • We then formed the rational expression: \[ \frac{g(2)}{f(2)} = \frac{6}{-5} \]
  • Simplifying, we get: \[ \frac{6}{-5} = -\frac{6}{5} \]
Rational expressions can often be simplified, and it鈥檚 essential to understand how to manipulate and simplify these expressions properly. In our exercise, we divided the results directly to get the solution.

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Most popular questions from this chapter

Solve. Write the answers using scientific notation. Without performing actual computations, explain why \(3^{-29}\) is smaller than \(2^{-29}\)

Simplify. \(\left(3 x^{2}\right)^{3}+4 x^{2} \cdot 4 x^{4}-x^{4}(2 x)^{2}+\left((2 x)^{2}\right)^{3}-\) \(100 x^{2}\left(x^{2}\right)^{2}\)

Find the domain of \(F / G,\) if $$ F(x)=\frac{1}{x-4} \quad \text { and } \quad G(x)=\frac{x^{2}-4}{x-3} $$

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Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)
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