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Theorem 1: Inverse Function
THEOREM 1 f has an inverse function if and only if f is onetoone. PROOF The following statements are equivalent. (1) f is a onetoone function. (2) For every y, either there is exactly one x with f(x) = y or there is no x with f(x) = y. (3) The equation y = f(x) determines x as a function of y. (4) f has an inverse function. COROLLARY Every function which is increasing on its domain I has an inverse function. So does every function decreasing on its domain I. PROOF Let f be increasing on I. For any two points x_{1} ≠ x_{2} in I, the value of f at the smaller of x_{1}, x_{2} is less than the value of f at the greater, so f(x_{1}) ≠ f(x_{2}). For example, the function y = x^{2} is not onetoone because (1)^{2} = 1^{2}, whence it has no inverse function. The function y = x^{2}, x ≥ 0, is increasing on its domain [0, ∞) and thus has an inverse. Now let us examine the trigonometric functions. The function y = sin x is not onetoone. For example, sin 0 = 0, sin π = 0, sin 2π = 0, etc. We can see in Figure 7.3.3 that the inverse relation of y = sin x is not a function. Figure 7.3.3 However, the function y = sin x is increasing on the interval [  π/2, π/2], because its derivative cos x is ≥ 0. So the sine function restricted to the interval [π/2, π/2], y = sin x,  π/2 ≤ x ≤ π/2, has an inverse function shown in Figure 7.3.4. This inverse is called the arcsine function. It is written x = arcsin y. Verbally, arcsin y is the angle x between  π/2 and π/2 whose sine is y. Figure 7.3.4


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