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Chapter 1 | Functions and Graphs
• For acute angles θ , the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is θ . • For a general angle θ , let ( x , y ) be a point on a circle of radius r corresponding to this angle θ . The trigonometric functions can be written as ratios involving x , y , and r . • The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period 2 π . The tangent and cotangent functions have period π . 1.4 Inverse Functions • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test. • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain. • For a function f and its inverse f −1 , f ⎛ ⎝ f −1 ( x ) ⎞ ⎠ = x for all x in the domain of f −1 and f −1 ⎛ ⎝ f ( x ) ⎞ ⎠ = x for all x in the domain of f . • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions. • The graph of a function f and its inverse f −1 are symmetric about the line y = x . 1.5 Exponential and Logarithmic Functions • The exponential function y = b x is increasing if b >1 and decreasing if 0< b <1. Its domain is (−∞, ∞) and its range is (0, ∞). • The logarithmic function y = log b ( x ) is the inverse of y = b x . Its domain is (0, ∞) and its range is (−∞, ∞). • The natural exponential function is y = e x and the natural logarithmic function is y = ln x = log e x . • Given an exponential function or logarithmic function in base a , we can make a change of base to convert this function to any base b >0, b ≠1. We typically convert to base e . • The hyperbolic functions involve combinations of the exponential functions e x and e − x . As a result, the inverse hyperbolic functions involve the natural logarithm.
CHAPTER 1 REVIEW EXERCISES True or False ? Justify your answer with a proof or a counterexample. 310. A function is always one-to-one.
f = x 2 +2 x −3,
g = ln( x −5),
h = 1
x +4
314. h
311. f ∘ g = g ∘ f , assuming f and g are functions.
315. g
316. h ∘ f
312. A relation that passes the horizontal and vertical line tests is a one-to-one function.
317. g ∘ f
313. A relation passing the horizontal line test is a function.
Find the degree, y -intercept, and zeros for the following polynomial functions. 318. f ( x ) =2 x 2 +9 x −5
For the following problems, state the domain and range of the given functions:
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