Example

What are the domain and range of the real-valued function [latex]f(x)=−3x^{2}+6x+1[/latex]?

Answer: This is a quadratic function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output. Because the coefficient of [latex]x^{2}[/latex] is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value. The vertex, or high point, is at ([latex]1, 4[/latex]). From the graph, you can see that [latex]f(x)\leq4[/latex].

Answer

The domain is all real numbers, and the range is all real numbers f(x) such that [latex]f(x)\leq4[/latex]. You can check that the vertex is indeed at ([latex]1, 4[/latex]). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same y value. The vertex must lie on the line of reflection, because it’s the only point that does not have a mirror image! In the previous example, notice that when [latex]x=2[/latex] and when [latex]x=0[/latex], the function value is [latex]1[/latex]. (You can verify this by evaluating [latex]f(2)[/latex] and [latex]f(0)[/latex].) That is, both ([latex]2, 1[/latex]) and ([latex]0, 1[/latex]) are on the graph. The line of reflection here is [latex]x=1[/latex], so the vertex must be at the point [latex](1, f(1))[/latex] . Evaluating f(1) gives [latex]f(1)=4[/latex], so the vertex is at [latex](1, 4)[/latex].

Example

What is the domain and range of the real-valued function [latex]f(x)=-2+\sqrt{x+5}[/latex]?

Answer: This is a radical function. The domain of a radical function is any x value for which the radicand (the value under the radical sign) is not negative. That means [latex]x+5\geq0[/latex], so [latex]x\geq−5[/latex]. Since the square root must always be positive or [latex]0[/latex], [latex] \displaystyle \sqrt{x+5}\ge 0[/latex]. That means [latex] \displaystyle -2+\sqrt{x+5}\ge -2[/latex].  

Answer

The domain is all real numbers x where [latex]x\geq−5[/latex], and the range is all real numbers f(x) such that [latex]f(x)\geq−2[/latex].

Example

What is the domain of the real-valued function [latex] \displaystyle f(x)=\frac{3x}{x+2}[/latex]?

Answer: This is a rational function. The domain of a rational function is restricted where the denominator is [latex]0[/latex]. In this case, [latex]x+2[/latex] is the denominator, and this is [latex]0[/latex] only when [latex]x=−2[/latex].  

Answer

The domain is all real numbers except [latex]−2[/latex]