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Chapter 8: Problem 59

What is the graph of a function?

Short Answer

Expert verified
The graph of a function is a visual representation of the relation between every input and its corresponding output. It helps to understand the behavior of the function.

Step by step solution

01

Define a Function

Firstly, identifying what a function is becomes very crucial. A function in mathematics can be thought of as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
02

Understanding the Graph

The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. It is a visual representation of a function. For a function of two variables, its graph exists in three dimensions.
03

Plotting the Graph

Plotting a graph is about drawing the function on a graph paper. You take the input values (usually on the x-axis), find the corresponding output values (usually on the y-axis), and plot these as points on a graph. Join these points to form a curve or a line. Each point on this curve or line corresponds to an input-output pair of the function.
04

Interpreting the Graph

Interpreting a graph is to understand what the function is trying to convey. This may show the increasing or decreasing nature of the function, any symmetries, and periodicity, and the limit behavior of the function for large input values.

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Most popular questions from this chapter

The functions are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$f(x)=\frac{2 x+1}{x-3}$$

Let $$\begin{array}{l}f(x)=2 x-5 \\\g(x)=4 x-1 \\\h(x)=x^{2}+x+2\end{array}$$. Evaluate the indicated function without finding an equation for the function. $$(g \circ f)(0)$$

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

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and g are inverses. $$f(x)=\frac{1}{x}+2, \quad g(x)=\frac{1}{x-2}$$

Simplify: \(3\left(\frac{x-2}{3}\right)+2\)
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