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Chapter 5: Problem 13

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ f(-4) / g(-4) $$

Short Answer

Expert verified
\(\frac{13}{18}\)

Step by step solution

01

- Find f(-4)

To find the value of f(x) at x = -4, substitute -4 into the function f(x):\[f(x) = -3x + 1\]\[f(-4) = -3(-4) + 1 = 12 + 1 = 13\]
02

- Find g(-4)

To find the value of g(x) at x = -4, substitute -4 into the function g(x):\[g(x) = x^2 + 2\]\[g(-4) = (-4)^2 + 2 = 16 + 2 = 18\]
03

- Calculate f(-4) / g(-4)

Using the values from Step 1 and Step 2, divide f(-4) by g(-4):\[\frac{f(-4)}{g(-4)} = \frac{13}{18}\]

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Key Concepts

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

function notation
Function notation is a way to represent and evaluate functions. It uses symbols like \(f(x)\) or \(g(x)\) to describe the relationship between an input \(x\) and an output. The function \(f(x) = -3x + 1\) means that for any input \(x\), you can find the output by multiplying \(x\) by -3 and then adding 1. Likewise, \(g(x) = x^2 + 2\) represents a function where you square the input \(x\) and then add 2. Using function notation makes it easier to show how functions are used and evaluated.
substitution
Substitution is the process of replacing a variable in a function with a specific value. This allows you to find the output for particular inputs. For example, to find \(f(-4)\), you substitute -4 for \(x\) in \(f(x)\). Start with the function \(f(x) = -3x + 1\).
Replace \(x\) with -4: \[f(-4) = -3(-4) + 1\]
Calculate: \(-3\) times \(-4\) is 12, and adding 1 gives 13, so \(f(-4) = 13\).
Similarly, for \(g(-4)\), use the function \(g(x) = x^2 + 2\), replace \(x\) with -4:
\[g(-4) = (-4)^2 + 2\]
Calculating it, \((-4)^2\) is 16, and adding 2 gives 18, so \(g(-4) = 18\).
division of functions
When you have two functions, you can divide one by the other, given that the denominator is not zero. In our example, divide \(f(x)\) by \(g(x)\) for \(x = -4\).
First, use the substituted values from before: \[f(-4) = 13\] and \[g(-4) = 18\].
Then, compute the division: \[\frac{f(-4)}{g(-4)} = \frac{13}{18}\].
This means that the value of \(\frac{f(-4)}{g(-4)}\) is approximately \(0.722\). Division of functions is useful in various mathematical analyses and applications, especially in calculus and algebra.

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