Example: Describing Sets on the Real-Number Line

Answer: To describe the values, [latex]x[/latex], included in the intervals shown, we would say, " [latex]x[/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5."

Example: Finding the Domain of a Function

Answer: The input value, shown by the variable [latex]x[/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers. In interval form, the domain of [latex]f[/latex] is [latex]\left(-\infty ,\infty \right)[/latex].

Example: Finding the Domain of a Function with an Even Root

Answer: When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for [latex]x[/latex].

[latex]\begin{cases}7-x\ge 0\hfill \\ -x\ge -7\hfill \\ x\le 7\hfill \end{cases}[/latex]

Example: Finding the Domain and Range Using Toolkit Functions

Answer: There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result. The domain is [latex]\left(-\infty ,\infty \right)[/latex] and the range is also [latex]\left(-\infty ,\infty \right)[/latex].

Example: Finding the Domain and Range

Answer: We cannot evaluate the function at [latex]-1[/latex] because division by zero is undefined. The domain is [latex]\left(-\infty ,-1\right)\cup \left(-1,\infty \right)[/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right)[/latex].

Example: Finding the Domain and Range

Answer: We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. [latex-display]x+4\ge 0\text{ when }x\ge -4[/latex-display] The domain of [latex]f\left(x\right)[/latex] is [latex]\left[-4,\infty \right)[/latex]. We then find the range. We know that [latex]f\left(-4\right)=0[/latex], and the function value increases as [latex]x[/latex] increases without any upper limit. We conclude that the range of [latex]f[/latex] is [latex]\left[0,\infty \right)[/latex].

Analysis of the Solution

Example: Finding Domain and Range from a Graph

Answer: We can observe that the horizontal extent of the graph is –3 to 1, so the domain of [latex]f[/latex] is [latex]\left(-3,1\right][/latex]. The vertical extent of the graph is 0 to [latex]–4[/latex], so the range is [latex]\left[-4,0\right)[/latex].

Example: Finding Domain and Range from a Graph of Oil Production

(credit: modification of work by the U.S. Energy Information Administration)

Answer: The input quantity along the horizontal axis is "years," which we represent with the variable [latex]t[/latex] for time. The output quantity is "thousands of barrels of oil per day," which we represent with the variable [latex]b[/latex] for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]1973\le t\le 2008[/latex] and the range as approximately [latex]180\le b\le 2010[/latex]. In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Example: Graphing a Piecewise Function

[latex]f\left(x\right)=\begin{cases}{ x }^{2} \text{ if }{ x }\le{ 1 }\\ { 3 } \text{ if } { 1 }<{ x }\le 2\\ { x } \text{ if }{ x }>{ 2 }\end{cases}[/latex]

Answer: Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality. Below are the three components of the piecewise function graphed on separate coordinate systems. (a) [latex]f\left(x\right)={x}^{2}\text{ if }x\le 1[/latex]; (b) [latex]f\left(x\right)=3\text{ if 1< }x\le 2[/latex]; (c) [latex]f\left(x\right)=x\text{ if }x>2[/latex] Now that we have sketched each piece individually, we combine them in the same coordinate plane.

Analysis of the Solution