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Inverse functions

Inverse functions. Do all functions have an inverse? Only functions that are monotonic (always increasing or decreasing) have inverses. In other words, only functions that are one-to-one (have no repeated y-values) have inverses.

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Inverse functions

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  1. Inverse functions • Do all functions have an inverse? • Only functions that are monotonic (always increasing or decreasing) have inverses. • In other words, only functions that are one-to-one (have no repeated y-values) have inverses. • In other words, only functions that pass the horizontal line test have inverses. 4.7 Inverse Trigonometric Functions

  2. What is an inverse function? • Recall the inverse of the exponential function • Logarithmic function • In general, how do we find the inverse of a given function? • Determine whether it has an inverse. • Swap the x and y variables and solve for y. 4.7 Inverse Trigonometric Functions

  3. Inverse trig functions • Do the trig functions have inverses? • Not at first, we have to restrict the domain. • Once we restrict the domain, what are the inverses of the trig functions? • y = sin(x) what is the x-input, y-output? • x = sin(y) swap the variables • arcsin(x) = arcsin(sin(y)) • arcsin(x) = y • or sin-1(x) = yinverse trig notation • The angle whose sine is x • x is the ratio of two sides of a triangle. • arcsin(x) = y is equivalent to sin(y) = x … why? 4.7 Inverse Trigonometric Functions

  4. WARNING: • sin-1(x) does not equal 1/sin(x) • Why? • The -1 denotes inverse notation • Just like f-1(x) does not denote reciprocal which would instead be • sin(x)-1 or 1/sin(x) 4.7 Inverse Trigonometric Functions

  5. Try it 4.7 Inverse Trigonometric Functions

  6. 4.7 Inverse Trig Functions Objectives: Identify the domain and range of the inverse trigonometric functions Use inverse trig functions to find angles Evaluate combinations of trig functions

  7. Calculator Practice • Use your calculator to evaluate the three problems from earlier • Do the calculator answers match your answers from the unit circle? 4.7 Inverse Trigonometric Functions

  8. WHY? Inverse Sine Function: Arcsin • If siny = x, then y = arcsinx • Sometimes labeled y=sin-1x (inverse notation) • -1≤x ≤1 • -π/2 ≤y ≤ π/2 • The domain is [-1,1] • The range is [-π/2, π/2] 4.7 Inverse Trigonometric Functions

  9. WHY? Inverse Cosine Function: Arccos • If cos(y) = x, then y = arccos(x) • Sometimes labeled y=cos-1x (inverse notation) • -1≤x ≤1 • 0 ≤y ≤ π • The domain is [-1,1] • The range is [0, π] 4.7 Inverse Trigonometric Functions

  10. WHY? Inverse Tangent Function: Arctan • If tany = x, then y = arctanx • Sometimes labeled y=tan-1x (inverse notation) • -∞< x <∞ • - π/2< y <π/2 • The domain is (-∞,∞) • The range is (- π/2, π/2) 4.7 Inverse Trigonometric Functions

  11. Example • Find the exact value. 4.7 Inverse Trigonometric Functions

  12. Inverse Properties: Example • Find the exact value (when possible) 4.7 Inverse Trigonometric Functions

  13. y  x Composition: Example • Find the exact value of 4.7 Inverse Trigonometric Functions

  14. y x  Composition: Example • Find the exact value of 4.7 Inverse Trigonometric Functions

  15. Evaluating inverse trig functions with a calculator • By definition, the values of inverse functions are always in radians. • arctan(-8.45) • sin-1(0.2447) • arccos(2) • sec-1(2) • arccsc(1) • cot-1(3) 4.7 Inverse Trigonometric Functions

  16. Closure Explain why the domain is [-1,1], and the range is [0, π] for the arccos function 4.7 Inverse Trigonometric Functions

  17. Assignments 4.7 Inverse Trigonometric Functions

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