Finding inverses Find the inverse function.


ƒ(x) = 3x² + 1, for x ≤ 0

Verified step by step guidance

1. 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.
2. Step 2: Replace \( f(x) \) with \( y \) to make it easier to work with: \( y = 3x^2 + 1 \).
3. Step 3: Solve for \( x \) in terms of \( y \). Start by isolating the \( x^2 \) term: \( y - 1 = 3x^2 \).
4. Step 4: Divide both sides by 3 to further isolate \( x^2 \): \( \frac{y - 1}{3} = x^2 \).
5. 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, \( f^{-1}(y) \).

Key Concepts

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.