Functions from a graphical perspective

Section1.13Functions from a graphical perspective

Subsection1.13.1Written Examples

1Determining function values and solving equations for functions presented in graphical form

Recall that a function is a set of ordered pairs with the property that no two points in the set share the same first coordinate. Specifically, if the function is a set of points of form \((x,y)\text{,}\) we say that \(y\) is a function of \(x\text{,}\) and if the name of the function is \(f\text{,}\) we write \(y=f(x)\text{.}\) The symbols \(y=f(x)\) are read aloud as "\(y\) equals \(f\) of \(x\)" or "\(y\) equals \(f\) at \(x\text{.}\)" The \(y\)-coordinate of he point is called the function value at the \(x\)-coordinate of the point.

Consider the function, \(f\text{,}\) that consists of the ordered pairs shown in Table 1.13.1. From the given ordered we know each of the following function values.

\begin{align*} f(0)\amp=7\\ f(1)\amp=-3\\ f(2)\amp=11\\ f(3)\amp=7 \end{align*}

\(x\)	\(y\)
\(0\)	\(7\)
\(1\)	\(-3\)
\(2\)	\(11\)
\(3\)	\(7\)

Table1.13.1The ordered pairs in \(f\)

Using words, we say that the value of \(f\) at \(0\) is \(7\text{,}\) the value of \(f\) at \(1\) is \(-3\text{,}\) the value of \(f\) at \(2\) is \(11\text{,}\) and the value of \(f\) at \(3\) is \(7\text{.}\)

When \(y\) is a function of \(x\text{,}\) the set of all \(x\)-coordinates in the set is called the domain of the function. When the domain of the function is a small, finite set, we can efficiently communicate the ordered pairs in the set via a table. However, when the domain is large, or infinite, tables are at best inefficient, and at worse totally ineffective vehicles for communicating the ordered pairs in the set. Two vehicles much more suitable in those cases are formulas and graphs. In this workshop we focus on graphs. There is another workshop where the focus is on formulas.

Consider the function, \(g\text{,}\) shown in Figure 1.13.2. From the points (ordered pairs) that have been stated, we can infer the following function values.

\begin{align*} g(-4)\amp=2\\ g(1)\amp=3\\ g(5)\amp=-1 \end{align*}

Figure1.13.2\(y=g(x)\)

If we were asked to evaluate the function at \(4\text{,}\) we would first locate the point with an \(x\)-coordinate of \(4\text{.}\) Then the function value would be the corresponding \(y\)-coordinate. In this case the relevant point is \((4,0)\) and we would report the function value by writing \(f(4)=0\text{.}\)

Now suppose that we were asked to solve the equation \(g(x)=4\text{.}\) Just like any other equation with the variable \(x\text{,}\) we are looking for all values that we could substitute for \(x\) that would result in a true statement. In words, the equation \(g(x)=4\) could be stated as "determine the values of \(x\) for which the value of the function\(g\) is \(2\text{.}\)" Since the function value corresponds the to the \(y\)-coordinate, we can further refine the charge as follows: "locate the points on the curve \(y=g(x)\) with \(y\)-coordinates equal to \(4\text{,}\) then the solutions to \(g(x)=4\) are the \(x\)-coordinates of those points"

In Figure 1.13.3 I've added the line \(y=4\) to a graph of \(y=g(x)\text{.}\) We can see that the line intersects the function \(g\) at the points \((-3,4)\) and \((0,4)\text{,}\) so the solutions to the equation \(g(x)=4\) are \(-3\) and \(4\text{.}\) Similarly, if we drew the line \(y=1\text{,}\) we see that is would intersect \(g\) at the points \((-4.5,1)\) and \((3,1)\text{,}\) so the solutions to the equation \(g(x)=1\) are \(-4.5\) and \(1\text{.}\)

Figure1.13.3\(y=g(x)\)

As a final example, let's consider the equation \(g(x)=-8\text{.}\) We can't draw the line \(y=-8\) onto Figure 1.13.3, nor can we see a point with an \(y\)-coordinate of \(10\text{,}\)so we'll have to take a different tact. The arrow on the left side of \(g\) implies that \(g\) continues forever in the leftward direction with the same slope. So one strategy we can employ is to use that slope to make a table for \(g\) until we find the point that has a \(y\)-coordinate of \(-8\text{.}\) I've done that in Table 1.13.4, and we can see that the relevant ordered pair is \((-8,-9)\text{.}\) From this we know that the solution to the equation \(g(x)=-8\) is \(-9\text{.}\)

\(x\)	\(y\)
\(-4\)	\(2\)
\(-5\)	\(0\)
\(-6\)	\(-2\)
\(-7\)	\(-4\)
\(-8\)	\(-6\)
\(-9\)	\(-8\)

Table1.13.4\(y=g(x)\)

2Solving inequalities involving functions presented in graphical form

When solving inequalities involving functions that are presented in graphical form, we follow a two-step process. We first use the function to identify the points on the curve whose \(-y\)-coordinates satisfy the property implied by the inequality statement. We then identify the \(x\)-coordinates of those points, which collectively make up the solution set to the inequality.

In Figure 1.13.5, I've indicated all of the points on the function named \(g\) that have \(y\)-coordinates greater than or equal to \(1\text{.}\) I've also marked off the portion of the \(x\)-axis over which these points lie. Since the \(y\)-coordinates of the points are the values of \(g(x)\text{,}\) we can infer from this that the solution set to the inequality \(g(x) \geq 1\) is \([-4.5,3]\text{.}\)

Figure1.13.5\(y=g(x)\)

In Figure 1.13.6, I've indicated all of the points on the function named \(g\) that have \(y\)-coordinates less than to \(44\text{.}\) I've also marked off the portion of the \(x\)-axis over which these points lie. Since the \(y\)-coordinates of the points are the values of \(g(x)\text{,}\) we can infer from this that the solution set to the inequality \(g(x) \lt 4\) is \((-\infty,-3) \cup (0,6]\text{.}\)

Figure1.13.6\(y=g(x)\)

3Determining the domain and range for functions presented in graphical form

Subsection1.13.2Practice Exercises (with step-by-step solutions)

1Determining function values and solving equations for functions presented in graphical form

Problem Set 1

Determine each of the following function values based upon the function \(f\) shown in Figure 1.13.7.

\(f(4)\)
\(f(2)\)
\(f(5)\)
\(f(1)\)

Determine the solution set to each of the following equations based upon the function \(f\) shown in Figure 1.13.7.

\(f(x)=-1\)
\(f(x)=2\)
\(f(x)=-3\)
\(f(x)=5\)

Problem Set 2

Determine each of the following function values based upon the function \(g\) shown in Figure 1.13.8.

\(g(-1)\)
\(g(2)\)
\(g(6)\)
\(g(-6)\)

Determine the solution set to each of the following equations based upon the function \(g\) shown in Figure 1.13.8.

\(g(x)=-2\)
\(g(x)=3\)
\(g(x)=-6\)
\(g(x)=-10\)

Problem Set 3

Determine each of the following function values based upon the function \(k\) shown in Figure 1.13.9.

\(k(2)\)
\(k(0)\)
\(k(4)\)
\(k(-3)\)

Determine the solution set to each of the following equations based upon the function \(k\) shown in Figure 1.13.9.

\(k(x)=1\)
\(k(x)=4\)
\(k(x)=5\)
\(k(x)=6\)

Solution

Problem Set 1

\(f(4)=2\)
\(f(2)=2\)
\(f(5)\) is not defined.
\(f(1)=5\)

The solutions set is \(\{-1,3\}\text{.}\)
The solution set is \(\{-4,-2,0,2,4\}\text{.}\)
The solution set is \(\emptyset\text{.}\)
The solution set is \(\{-3,1\}\)

Problem Set 2

\(g(-1)=6\)
\(g(2)=0\)
\(g(6)=-8\)
\(g(-6)\) is not defined.

The solution set is \(\{-2,3\}\text{.}\)
The solution set is \(\{\frac{1}{2}\}\text{.}\)
The solution set is \(\{-4,5\}\)
The solution set is \(\{7\}\text{.}\)

Problem Set 3

\(k(2)\) is undefined.
\(k(0)-3\)
\(k(4)=4\)
\(k(-3)=-5\)

The solution set is \(\{-1,5\}\text{.}\)
The solution set is \(\{4\}\text{.}\)
The solution set is \(\{3\}\text{.}\)
The solution set is \(\emptyset\text{.}\)

2Solving inequalities involving functions presented in graphical form

Problem Set 1

Determine the solution set to each of the following equations based upon the function \(f\) shown in Figure 1.13.10. State the solution set using both set-builder notation and interval notation.

\(f(x) \lt 2\)
\(f(x) \geq 2\)
\(f(x) \lt -1\)
\(f(x) \leq 5\)

Problem Set 2

Determine the solution set to each of the following inequalities based upon the function \(g\) shown in Figure 1.13.11. State the solution set using both set-builder notation and interval notation.

\(g(x) \geq -2\)
\(g(x) \leq 3\)
\(g(x) \lt -6\)
\(g(x) \gt -6\)

Problem Set 3

Determine the solution set to each of the following inequalities based upon the function \(k\) shown in Figure 1.13.12. State the solution set using both set-builder notation and interval notation.

\(k(x) \geq 1\)
\(k(x) \leq 4\)
\(k(x) \gt 5\)
\(k(x) \lt 6\)

Solution

Problem Set 1

The solution set is \(\{x \mid -2 \lt x \lt 0 \text{ or } 2 \lt x \lt 4\}\text{.}\)

The solution set is \((-2,0) \cup (2,4)\text{.}\)
The solution set is \(\{x \mid -4 \leq x \leq -2 \text{ or } 0 \leq x \leq 2 \text{ or } 4 \leq x \lt 5\}\text{.}\)

The solution set is \([-4,-2] \cup [0,2] \cup [4,6)\text{.}\)
The solution set is \(\{\}\text{.}\)

The solution set is \(\emptyset\text{.}\)
The solution set is \(\{x \mid -4 \leq x \lt 5\}\text{.}\)

The solution set is \([-4,5)\text{.}\)

Problem Set 2

The solution set is \(\{-2 \leq x \leq 3\}\text{.}\)

The solution set is \([-2,3]\text{.}\)
The solution set is \(\{x \mid -6 \lt x \lt 2 \text{ or } x \geq \frac{1}{2}\}\text{.}\)

The solution set is \((-6,-2) \cup [\frac{1}{2},\infty)\text{.}\)
The solution set is \(x \mid x \gt 5\}\text{.}\)

The solution set is \((5,\infty)\text{.}\)
The solution set is \(\{x \mid -6 \lt x \lt -4 \text{ or } -4 \lt x \lt 5\}\text{.}\)

The solution set is \((-6,-4) \cup (-4,5)\text{.}\)

Problem Set 3

The solution set is \(\{x \mid -1 \leq x \lt 0 \text{ or } 2 \lt x \leq 5\}\text{.}\)

The solution set is \([-1,0) \cup (2,5]\text{.}\)
The solution set is \(\{x \mid x \leq 0 \text{ or } x \geq 4\}\text{.}\)

The solution set is \((-\infty,0] \cup [4,\infty)\text{.}\)
The solution set is \(\{\}\text{.}\)

The solution set is \(\emptyset\text{.}\)
The solution set is \(\{x \mid x \leq 0 \text{ or } x \gt 2\}\text{.}\)

The solution set is \((-\infty,0] \cup (2,\infty)\text{.}\)

3Determining the domain and range for functions presented in graphical form

Problem 1

Determine the domain and range of the function \(f\) shown in Figure 1.13.13. State the domain and range using interval notation.

Problem 2

Determine the domain and range of the function \(g\) shown in Figure 1.13.14. State the domain and range using interval notation.

Problem 3

Determine the domain and range of the function \(k\) shown in Figure 1.13.18. State the domain and range using interval notation.

Solution

Problem 1

The domain is \([-4,5),\text{.}\) The range is \([-1,5]\text{.}\)

Problem 2

The domain is \((-6,\infty)\text{.}\) The range is \((-\infty,6]\text{.}\)

Problem 3

The domain is \((-\infty,0] \cup (2,\infty)\text{.}\) The range is \((-\infty,6]\text{.}\)