Use the graph of ƒ to find ƒ⁻¹ (2),ƒ⁻¹ (9), and ƒ⁻¹ (12) <IMAGE>

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Step 1: Understand the concept of an inverse function. The inverse function \( f^{-1} \) of a function \( f \) is a function that reverses the effect of \( f \). If \( f(a) = b \), then \( f^{-1}(b) = a \).

Step 2: Identify the values of \( y \) for which you need to find the inverse: \( f^{-1}(2) \), \( f^{-1}(9) \), and \( f^{-1}(12) \). This means you need to find the \( x \)-values such that \( f(x) = 2 \), \( f(x) = 9 \), and \( f(x) = 12 \).

Step 3: Examine the graph of \( f \) to locate the points where the function value (\( y \)-value) is 2, 9, and 12. For each \( y \)-value, find the corresponding \( x \)-value on the graph.

Step 4: For each \( y \)-value, note the \( x \)-coordinate of the point on the graph where the function value is equal to that \( y \)-value. This \( x \)-coordinate is the value of the inverse function at that \( y \)-value.

Step 5: Write down the results: \( f^{-1}(2) \), \( f^{-1}(9) \), and \( f^{-1}(12) \) as the \( x \)-values you found in the previous step.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function reverses the effect of the original function. If a function f takes an input x and produces an output y, then its inverse f⁻¹ takes y as input and returns x. Understanding how to find inverse functions is crucial for solving problems that require determining specific outputs from given inputs.

Graph Interpretation

Interpreting graphs involves analyzing the visual representation of functions to extract information about their behavior. For inverse functions, this means identifying points on the graph where the output corresponds to the desired input values. This skill is essential for accurately finding values of f⁻¹ from the graph of f.

Function Notation

Function notation is a way to denote functions and their inverses clearly. For example, f(x) represents the output of function f for input x, while f⁻¹(y) denotes the input that produces output y in the inverse function. Familiarity with this notation is important for correctly interpreting and solving problems involving functions and their inverses.

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Textbook Question

Solving equations Solve each equation.

sin² 2Θ = 1/2, -π/2 ≤ Θ ≤ π/2

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Textbook Question

Defining piecewise functions Write a definition of the function whose graph is given <IMAGE>

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