This is a quadratic function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing to restrict from 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].

The domain is all real numbers, and the range is all real numbers [latex]f(x)[/latex] such that [latex]f(x)\leq4[/latex].

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].

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

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].

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