AP Precalculus Section 3.9 Example: Find the Inverse Function of g(t)=-1/3sin(2t-π/2)+5

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

To find the inverse function of \( g(t) = -\frac{1}{3}\sin(2t - \frac{\pi}{2}) + 5 \), follow these steps:

1. **Write the Function in Terms of \( y \):**
- Replace \( g(t) \) with \( y \): \( y = -\frac{1}{3}\sin(2t - \frac{\pi}{2}) + 5 \).

2. **Swap \( t \) and \( y \):**
- Swap \( t \) and \( y \): \( t = -\frac{1}{3}\sin(2y - \frac{\pi}{2}) + 5 \).

3. **Solve for \( y \):**
- Solve the equation for \( y \):
\[ \sin(2y - \frac{\pi}{2}) = \frac{3}{2}(5 - t) \]
\[ 2y - \frac{\pi}{2} = \arcsin\left(\frac{3}{2}(5 - t)\right) \]
\[ 2y = \arcsin\left(\frac{3}{2}(5 - t)\right) + \frac{\pi}{2} \]
\[ y = \frac{1}{2}\left(\arcsin\left(\frac{3}{2}(5 - t)\right) + \frac{\pi}{2}\right) \]

4. **Replace \( y \) with \( g^{-1}(t) \):**
- Replace \( y \) with \( g^{-1}(t) \):
\[ g^{-1}(t) = \frac{1}{2}\left(\arcsin\left(\frac{3}{2}(5 - t)\right) + \frac{\pi}{2}\right) \]

So, the inverse function of \( g(t) = -\frac{1}{3}\sin(2t - \frac{\pi}{2}) + 5 \) is \( g^{-1}(t) = \frac{1}{2}\left(\arcsin\left(\frac{3}{2}(5 - t)\right) + \frac{\pi}{2}\right) \).

Make sure to consider the domain of \( g(t) \) to ensure that the inverse function is well-defined. In this case, the domain of \( g(t) \) should be limited to ensure that the inverse function exists.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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