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Chapter 1 | Functions and Graphs
The domain and range of f −1 are given by the range and domain of f , respectively. Therefore, the domain of f −1 is [0, ∞) and the range of f −1 is [−1, ∞). To find a formula for f −1 , solve the equation y = ( x +1) 2 for x . If y = ( x +1) 2 , then x =−1± y . Since we are restricting the domain to the interval where x ≥−1, we need ± y ≥0. Therefore, x =−1+ y . Interchanging x and y , we write y =−1+ x and conclude that f −1 ( x ) =−1+ x .
1.26 Consider f ( x ) =1/ x 2 restricted to the domain (−∞, 0). Verify that f is one-to-one on this domain. Determine the domain and range of the inverse of f and find a formula for f −1 .
Inverse Trigonometric Functions The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function ( Figure 1.34 ). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval ⎡ ⎣ − π 2 , π 2 ⎤ ⎦ . By doing so, we define the inverse sine function on the domain [−1, 1] such that for any x in the interval [−1, 1], the inverse sine function tells us which angle θ in the interval ⎡ ⎣ − π 2 , π 2 ⎤ ⎦ satisfies sin θ = x . Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions , which are functions that tell us which angle in a certain interval has a specified trigonometric value. Definition The inverse sine function, denoted sin −1 or arcsin, and the inverse cosine function, denoted cos −1 or arccos, are defined on the domain D ={ x | −1≤ x ≤1} as follows:
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