SECTION 2.1 Functions 69 Solution ↑ ( ) ( ) ( ) + = + − + f x h x h x h 2 3 2 (a) [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) + − = + − + − − = + + − − − + = + + − − = + − = + − = + − f x h f x h x h x h x x h x xh h x h x x h x xh h h x h xh h h h h x h h x h 2 3 2 3 2 2 3 3 2 3 2 4 2 3 2 4 2 3 4 2 3 4 2 3 2 2 2 2 2 2 2 2 2 (b) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − = + − = − + + = − − + = − + = − + f x h f x h x h x h x x h x x h h x x h x x h h h x x h h x x h 4 4 4 4 4 4 4 4 4 (c) When a function involves a square root, we rationalize the numerator of expression (1). After simplifying, the h’s are divided out. ( ) ( ) ( ) ( ) ( ) ( ) + − = + − = + − ⋅ + + + + = + − + + = + + = + + f x h f x h x h x h x h x h x h x x h x x h x h x h x h h x h x x h x 1 2 2 Simplify. Distribute and combine like terms. Combine like terms. Factor out h. Divide out the h’s. ( ) + = + f x h x h 4 Subtract. Divide and distribute. Simplify. Divide out the h’s. ( ) + = + f x h x h Rationalize the numerator. ( )( ) − + = − A B A B A B 2 2 ( ) ( ) + − = + − = x h x x h x h 2 2 Divide out the h’s. Now Work PROBLEM 83 5 Find the Domain of a Function Defined by an Equation Often the domain of a function f is not specified; instead, only the equation defining the function is given. In such cases, the domain of f is the largest set of real numbers for which the value ( ) f x is a real number. The domain of a function f is the same as the domain of the variable x in the expression ( ) f x . Need to Review? Rationalizing Numerators is discussed in Section A.10, pp. A89–A90.
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