Functions - Domain and Range; Composition
Sol 1 f is defined for all values of x (since f is a polynomial), so the domain of f is . Since the graph of f is a parabola which opens downward with vertex at , the set of y-coordinates for the points on the graph of f consists of all y-values with ; so the range of f is the interval .
Sol 2 f is defined where or , so the domain of f is the interval . Since , for any x in the domain of f; so the range of f is contained in . If , (since ), so the range of f is actually equal to .
Sol 3 f is defined for or , so the domain of f is given by . To find the range of f, we must determine for which y-values the equation has a solution for x. Multiplying both sides of this equation by gives or , so . Therefore the equation has a solution for x iff or , so the range of f is given by .
Sol 4 f is defined wherever (so the square root is defined) and (so the fraction is defined). Solving the inequality or gives , so the domain of f is .
Sol 5 , while .
Sol 6 We can let and , for example.
Sol 7 Since , and therefore and .
Sol 8 is defined where , so gives . Taking the nonnegative square root of both sides gives or . Therefore is the domain of .
Sol 9 is defined where
Marking 0,5,3, and -3 on a number line and using the facts that all factors have odd exponents and that ,
we get the following sign chart for :
Therefore the domain of is given by
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