5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

Review of Precalculus 63

4. Solve fory.Thus,y=
   x^2 + 1
   2

.

5. Replaceybyf−^1 (x). You have f−^1 (x)=
   x^2 + 1
   2

.

6. Since the range off(x)is[0,∞), the domain of f−^1 (x)is[0,∞).


7. Verifyf−^1 (x) by checking:
   Sincex>0,

√

x^2 =x,

f(f−^1 (x))=f

(

x^2 + 1

2

)

=

√

2

(

x^2 + 1

2

)

− 1 =x

f−^1 (f(x))=f−^1 (

√

2 x−1)=

(

√

2 x−1)^2 + 1

2

=x

Trigonometric and Inverse Trigonometric Functions

There are six basic trigonometric functions and six inverse trigonometric functions. Their

graphs are illustrated in Figures 5.3-9 to 5.3-20.

− 2

− 1

0

y = sin x

Domain: {−∞ < x < ∞}

Range: {− 1 ≤ y ≤ 1}

Frequency: 1

Amplitude: 1

Period: 2π

x

y

− 2 π −1. 5π− 1 π −0.5π 0.5π 1 π 1. 5π 2 π

1

2

Figure 5.3-9

x

y

1 2

y = sin−^1 x

− 2 − 1 0

0.5π

−0.5π

Domain: {− 1 ≤ x ≤ 1}

Range: {− 2 π ≤ y ≤ 2 π}

Figure 5.3-10

− 2

− 1

0

y = cos x

x

y

− 2 π−1. 5π − 1 π −0.5π 0.5π 1 π 1. 5π 2 π

1

2

Domain: {−∞ < x < ∞}

Range: {− 1 ≤ y ≤ 1}

Frequency: 1

Amplitude: 1

Period: 2π

Figure 5.3-11

x

y

1 2

y = cos–^1 x

• 2 – 1 0

π

0.5π

Domain: {– 1 ≤ x ≤ 1}

Range: {0 ≤ y ≤ π}

Figure 5.3-12