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Chapter 2: Problem 113

What is the graph of a function?

Short Answer

Expert verified
The graph of a function is a visualization of the relationship between input and output variables of the function on a coordinate plane. It's the set of all points for which the x-coordinates are the inputs of the function and the y-coordinates are the outputs of the function. This helps us understand and visualize the behavior of the function.

Step by step solution

01

Understanding Function

A function is a special kind of relation in which each input is related to exactly one output. In terms of Mathematics, it's a relation between a set of inputs and a set of permissible outputs, whereby each input is related to exactly one output. For example, a function \(f(x) = x^2 + 3x + 2\). The 'x' term can represent different input values, and for each of those, the function will have a different output value.
02

Understanding Graph

A graph in mathematics is a tool that visualizes various types of relations between numbers. The graph is drawn on a grid called a coordinate plane. The horizontal line is called the x-axis and the vertical line is the y-axis. Every point on the graph can be represented by an ordered pair of numbers (x, y), called coordinates.
03

Connecting Function and Graph

The graph of a function is the set of all points in the x-y plane for which the x-coordinates are the inputs of the function and the y-coordinates are the outputs of the function. For instance, in the function \(f(x) = x^2 + 3x + 2\), you can substitute any 'x' values to get the corresponding 'y' values, and hence, plot the graph of the function. This helps in visualizing the behavior of the function.
04

Practical illustration

To better illustrate, let's take the function \(f(x) = x^2\). When we put the value of \(x = -2\), the output \(y = 4\). Similarly, when \(x = -1\), the output \(y = 1\). This way we can derive more points, plot them on the graph and then draw our graph for the function \(x^2\).

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Most popular questions from this chapter

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}-6 y-7=0 $$

will help you prepare for the material covered in the next section. $$ \text { If }\left(x_{1}, y_{1}\right)=(-3,1) \text { and }\left(x_{2}, y_{2}\right)=(-2,4), \text { find } \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$

The function $$ f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95 $$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a \([0,100,5]\) by \([0,40,2]\) viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the \([\mathrm{TABLE}]\) or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?

The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve Exercises \(101-102\) a. Find and interpret \(f(70)\) b. Find and interpret \(g(70)\) c. Which function serves as a better model for the chance of surviving to age \(70 ?\)

Exercises \(129-131\) will help you prepare for the material covered in the next section. Use point plotting to graph \(f(x)=x+2\) if \(x \leq 1\)
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