Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

Verified step by step guidance

Step 1: Understand the function and its domain. The function given is \( f(x) = 3x^2 + 1 \) with the domain \( x \leq 0 \). This means we are only considering the left half of the parabola.

Step 2: Replace \( f(x) \) with \( y \) to make it easier to work with: \( y = 3x^2 + 1 \).

Step 3: Solve for \( x \) in terms of \( y \). Start by isolating the \( x^2 \) term: \( y - 1 = 3x^2 \).

Step 4: Divide both sides by 3 to further isolate \( x^2 \): \( \frac{y - 1}{3} = x^2 \).

Step 5: Since \( x \leq 0 \), take the negative square root to solve for \( x \): \( x = -\sqrt{\frac{y - 1}{3}} \). This expression represents the inverse function, \( f^{-1}(y) \).

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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 essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). When finding an inverse function, it is crucial to consider the domain of the original function, as it affects the range of the inverse function. In this case, the restriction x ≤ 0 is important for determining the inverse.

Solving for x

To find the inverse of a function, one typically starts by replacing f(x) with y, then solving for x in terms of y. This often involves algebraic manipulation, such as isolating x on one side of the equation. Once x is expressed in terms of y, the inverse function can be written as f⁻¹(y) = x, and then it can be rewritten in terms of x.

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Textbook Question

Solving equations Solve each equation.

sin² 2Θ = 1/2, -π/2 ≤ Θ ≤ π/2

Finding inverses Find the inverse function.ƒ(x) = 3x² + 1, for x ≤ 0