Chapter 2.1, Problem 7AYU

To determine

To fill:The intervalused to define domain of the composite function $f+g$ .

Answer to Problem 7AYU

Domain of the composite function $f+g$ is all real numbers in the interval $\left[0,5\right]$ .

Explanation of Solution

Given information:

Domain of $f$ is all real numbers in the interval $\left[0,7\right]$ and domain of $g$ is all real numbers in the interval $\left[-2,5\right]$ .

Consider two sets A and B . If each element $x$ in the set A has exactly one element $y$ in the set B , the relation between them is known as a function. The ordered pair is $\left(x,y\right)$ .

The values of first coordinate $x$ is known as input values and the values of second coordinate $y$ is known as output values of a function.

Therefore, the set of input values represent the domain and set of output values represent the range of the function.

Hence, set of all images of the elements in the domain of a function is defined as the range of the function.

For a composite function $f+g$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

Now, $\text{Domain}\text{of}f=\left[0,7\right]$

And $\text{Domain}\text{of}g=\left[-2,5\right]$

So,

  $\begin{array}{c}\text{Domain}\text{of}f+g=\left[0,7\right]\cap \left[-2,5\right]\\ =\left[0,5\right]\end{array}$

Therefore, domain of the composite function $f+g$ is all real numbers in the interval $\left[0,5\right]$ .