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Explanation & Answer
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1.) What is the domain and range of y cos x ?
Domain: the set of all real numbers : (-∞, ∞)
Range: the set of all real numbers from -1 to 1, inclusive: [-1, 1]
2.) True or False: For a trigonometric function,
y f x
, then
x f 1 y
. Explain your answer.
False. Trigonometric functions are not one to one. Hence the inverse doesn’t exist. Inverse function
exists only for trigonometric functions with restricted domain.
3.) True or False: For a one-to-one function,
y f x
, then
x f 1 y
. Explain your answer.
True. Since the function is one to one, the inverse exists. By the definition of inverse, y = f(x), x = f-1(y).
4.) True or False: For any function,
x f 1 y
, then
y f x
. Explain your answer.
False, inverse exists only for one to one functions. All functions are not one to one.
3
,
y
sin
x
5.) True or False: The graph of
is increasing on the interval 2 2 . Explain your answer.
3
,
False. Sin(π/2) = 1, sin(π) = 0 and sin(3π/2) = -1. Therefore, the function is decreasing on 2 2
1
6.) Explain the meaning of y cos x .
y cos1 x means x = cos(y),
-1 ≤ x ≤ 1.
7.) True or False: sec 1 0.5 is undefined. Explain your answer.
1
True. 𝑠𝑒𝑐 −1 (0.5) = 𝑐𝑜𝑠 −1 ( ) = 𝑐𝑜𝑠 −1 (2), which is undefined since the domain of y = cos-1x is
0.5
[-1, 1].
8.) Find the exact value of each expression. Do not use a calculator.
9.)
Cos1 0
cos-1(0) = x, then cos(x) = 0,
x = π/2.
Hence cos-1(0) = π/2
0≤x≤π
10.) Sin1 1
Let sin-1(1) = x,
Then sin(x) = 1, -π/2 ≤ x ≤ π
x = π/2
hence sin-1(1) = π/2
3
11.) Tan1
3
3
Tan1
3
= x,
Let
Tan(x) =
𝑥 =
√3
3
−𝜋
,
2
<
𝜋
2
𝜋
6
√3
𝜋
3
6
𝐻𝑒𝑛𝑐𝑒 𝑡𝑎𝑛−1 ( ) =
Use a calculator to find the value of each expression. Round your answer to the nearest hundredth.
12.) Arcsin
...
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