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Chapter 8: Problem 36

Explain how to determine whether a relation is a function. What is a function?

Short Answer

Expert verified
A function is a relation between sets where each input has exactly one output. A relation is a function if each input from the first set (or each x-value) has only one corresponding output in the second set (or one y-value).

Step by step solution

01

Definition of a Function

A function is a rule that assigns each element in one set (called the domain) to exactly one element in another set (called the range). In simpler terms, it is a relation in which every input has exactly one output.
02

Relation

A relation is any set of ordered pairs. The first component of the ordered pairs is from the first set, and the second component is from the second set.
03

Determining if a Relation is a Function

Now, to determine if a relation is a function, check if each input from the first set (or each x-value) has only one corresponding output in the second set (or one y-value). If any input corresponds to two or more different outputs, then the relation is not a function. Using a vertical line test on a graph can also help: If a vertical line passes through more than one point on the graph, then it is not a function.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm working with the linear function \(f(x)=3 x+5\) and I do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\).

\(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{c|c}\hline x & f(x) \\\\\hline-1 & 1 \\\\\hline 0 & 4 \\\\\hline 1 & 5 \\\\\hline 2 & -1 \\ \hline\end{array}$$ $$\begin{array}{c|c}\hline x & g(x) \\\\\hline-1 & 0 \\\\\hline 1 & 1 \\\\\hline 4 & 2 \\\\\hline 10 & -1 \\ \hline\end{array}$$ $$f(g(1))$$

If \(f(x)=x^{2}+x\) and \(g(x)=x-5,\) find \(f(4)+g(4)\).

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=4 x+9 \quad \text { and } \quad g(x)=\frac{x-9}{4}$$

Explain how to determine if two functions are inverses of each other.
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