Question

In Exercises 49 and \(50,\) (a) draw the graph of the function. Then find its (b) domain and (c) range. $$f(x)=x+5, \quad g(x)=x^{2}-3$$

Short answer

The graphs of \(f(x)=x+5\) and \(g(x)=x^{2}-3\) are a straight line and a parabola respectively. Their domains are both \(R\), while the range of \(f(x)=x+5\) is \(R\) and the range of \(g(x)=x^{2}-3\) is \([-3, +\infty)\).

Step-by-step solution

Graph the function \(f(x)=x+5\)

First, plot the function \(f(x)=x+5\). As a simple linear function, it will look like a straight line. To sketch the graph, pick a few x-values, find the appropriate y-value by plugging the x-values into the function, and connect the dots to form a straight line.

Find the domain and range of \(f(x)=x+5\)

Looking at the graph, it can be seen that the function is defined for all real numbers, hence the domain is \(R\) (all real numbers). Since the line goes on infinitely in both the positive and negative y-direction, the range of the function is also \(R\).

Graph the function \(g(x)=x^{2}-3\)

For the function \(g(x)=x^{2}-3\), pick a few different x-values, square them, and then subtract 3. This function forms a parabola shaped graph.

Find the domain and range of \(g(x)=x^{2}-3\)

Again looking at the graph for \(g(x)=x^{2}-3\), it also covers all x-values or all real numbers so the domain is \(R\). Because it is a square function shifted downward by three units, the y-values of the function are greater than or equal to -3. Therefore, the range is \([-3, +\infty)\).