SAT Domain and Range: Identifying Function Restrictions and Intervals

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values). For a simple function like f(x)=x+1, the domain is all real numbers (you can plug in any x and get an output). The range is also all real numbers. But for f(x)=1/x, the domain excludes x=0 (division by zero is undefined), so domain is all real numbers except 0. For f(x)=sqrt(x), the domain includes only non-negative numbers (you cannot take the square root of a negative number in the real numbers), so domain is x>=0. The key restrictions on domain come from three sources: (1) division by zero, (2) square roots of negative numbers (for real-valued functions), and (3) logarithms of non-positive numbers; identifying these restrictions determines where a function is defined.

Range is trickier because it depends on the function's behavior. For f(x)=x^2, the domain is all real numbers, but the range is only y>=0 (the outputs are never negative because squaring always gives non-negative results). For f(x)=1/x, the range is all real numbers except 0 (as x approaches 0 from either side, f(x) approaches positive or negative infinity, but f(x) never equals 0). Finding range often requires sketching the graph or analyzing the function's behavior algebraically. On the SAT, range questions usually involve recognizing that certain output values are impossible due to the function's structure.

For rational functions (fractions with polynomials), exclude any x-value that makes the denominator zero. For f(x)=(x+2)/(x^2-4), factor the denominator: x^2-4=(x-2)(x+2). The function is undefined when x=2 or x=-2, so the domain is all real numbers except -2 and 2. For radical functions, exclude any x-value that makes the radicand (expression under the radical) negative. For f(x)=sqrt(3x-6), the radicand 3x-6 must be non-negative: 3x-6>=0, so x>=2. Domain is x>=2. For f(x)=sqrt(x^2-4), the radicand x^2-4>=0, which factors as (x-2)(x+2)>=0, so x<=-2 or x>=2. When finding domain restrictions from inequalities like x^2-4>=0, remember that squaring (or other powers) can reverse inequality direction, so test critical points to determine which regions satisfy the inequality. A quick check on a number line prevents sign errors.

A 3-step process for identifying domain from expressions: (1) Identify sources of restriction (denominators, radicals, logarithms). (2) Write restriction equations or inequalities (denominator≠0, radicand>=0). (3) Solve the inequalities and write domain in interval notation or set notation. For f(x)=sqrt(x+3)/(x-1), the numerator requires x+3>=0 (so x>=-3) and the denominator requires x-1≠0 (so x≠1). Combined domain: x>=-3 and x≠1, written as [-3,1)∪(1,∞).

For quadratic functions like f(x)=x^2-4x+3, complete the square or note the vertex to find range. Rewrite as f(x)=(x-2)^2-1. The vertex is at (2,-1), and the parabola opens upward, so the minimum value is -1. Range is y>=-1. For f(x)=-2(x+1)^2+5, the vertex is (-1,5), the parabola opens downward (negative leading coefficient), so maximum value is 5. Range is y<=5. For rational functions, sketch the graph or recognize asymptotes. f(x)=1/x has a horizontal asymptote at y=0 (as x approaches infinity, y approaches 0), so y≠0. For f(x)=(x+2)/(x-1), rewrite: y=(x+2)/(x-1) implies y(x-1)=x+2, so yx-y=x+2, rearranging: x(y-1)=y+2, so x=(y+2)/(y-1). This is undefined when y=1, so y≠1. To find range of a rational function algebraically, solve for x in terms of y and identify which y-values make the expression undefined; those y-values are excluded from the range.

For inverse functions, range of the original function equals domain of the inverse. If f(x)=x^2 has range [0,∞), then its inverse has domain [0,∞). This relationship provides another way to check your range answers: the domain of the inverse should match the range of the original.

When a question asks for domain or range, use a systematic approach. First, identify the type of function (linear, quadratic, rational, radical, etc.). Second, identify sources of restriction. Third, solve to find the restriction. Fourth, write the answer in the required format (interval notation, set notation, inequality, or a list of excluded values). Practice problems often ask, "What is the domain of f(x)=..." and expect an interval notation answer or a statement like "all real numbers except x=2." A checklist for domain problems: (1) Check for denominators—exclude values where denominators equal zero. (2) Check for radicals (even roots)—require non-negative radicand. (3) Check for logarithms—require positive argument. (4) Combine restrictions. (5) Write the domain. Applying this checklist to 10 practice problems per day for a week builds the routine so that domain problems feel straightforward rather than requiring thought on test day.

For range problems, you may need to sketch the graph or analyze the function's extreme values. If the function is quadratic, find the vertex. If rational, identify asymptotes. If it includes a radical, recognize that radical functions have restricted ranges (e.g., sqrt never outputs negative). Test-day strategy: if range is hard to determine algebraically, sketch the graph and read the range from the graph. This visual approach often works faster than algebraic analysis under time pressure.