Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=2 x^2-5 $$

Short answer

The inverse of the function \(f(x) = 2x^2 - 5\) is not a function unless we limit the domain to \(x ≥ -5\). The inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = + \sqrt{(x + 5) / 2}\) for \(x ≥ -5\).

Step-by-step solution

Check whether inverse is a function

To check if the inverse of \(f(x) = 2x^2 -5\) is a function, we need to see if the graph of \(f\) passes the 'horizontal line test'. This test states that a function \(f\) has an inverse which is also a function, if and only if every horizontal line intersects the graph of \(f\) at most once. Since \(f(x) = 2x^2 -5\) is a quadratic function that opens upwards, it fails the horizontal line test. Hence, the inverse of the function \(f(x)\) is not a function over all real numbers.

Find the inverse

To find the inverse, we replace the function \(f(x)\) with \(y\), interchange the roles of \(y\) and \(x\), and then solve for \(y\). Therefore, starting from the function \(f(x) = 2x^2 - 5\), we write it as \(y = 2x^2 - 5\). Interchanging \(x\) and \(y\) yields \(x = 2y^2 - 5\), and solving for \(y\) gives \(y = ± \sqrt{(x + 5) / 2}\). However, since the inverse must also be a function, we limit the domain. Thus, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = + \sqrt{(x + 5) / 2}\) for \(x ≥ -5\).

Key concepts

Inverse Functions

An inverse function essentially reverses the roles of inputs and outputs for a given function. If you have a function that maps an input to an output, the inverse function maps that output back to the input. For example, if a function turns 3 into 9, the inverse function will turn 9 back into 3. The notation for an inverse is often represented as \(f^{-1}(x)\). A crucial aspect of inverses is that not every function has an inverse that is also a function. A function must be both one-to-one and onto in its domain to have an inverse that is a function. This means that each output is achieved from exactly one input, and each input is used to produce exactly one output. For functions that do not meet this criterion, their inverses may not represent proper functions without certain restrictions.

Horizontal Line Test

The horizontal line test helps determine if a function's inverse is a functional relationship. Imagine drawing a series of horizontal lines across the graph of a function. If any line touches the graph more than once, the inverse of that function is not also a function. For the function \(f(x) = 2x^2 - 5\), a quadratic function, the graph is a parabola that opens upwards. If you test it with horizontal lines, you'll see they can intersect the curve at two points. This means the inverse would not pass the vertical line test needed for it to be a function. To resolve this, limitations are applied to decrease the domain so that the inverse reflects a true functional relationship.

Quadratic Functions

A quadratic function is any function that can be represented as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The graph of a quadratic function is a parabola. In the given function \(f(x) = 2x^2 - 5\), it opens upwards because the coefficient of \(x^2\) (which is 2) is positive. Quadratic functions can have their domains restricted to make their inverses functions, usually by considering only half of the parabola, resulting in a limit to one arm that doesn't intersect horizontal lines more than once. This restriction is often necessary when an inverse is needed for practical purposes or calculations in various applications.

Domain and Range

The domain of a function refers to all possible input values, while the range is the set of all possible output values. Both domain and range are crucial in understanding the behavior and constraints of a function and its inverse. In the initial function \(f(x) = 2x^2 - 5\), the domain is all real numbers because you can input any real number to produce a valid output. The range, on the other hand, is all real numbers greater than or equal to \(-5\). The inverse problem of this function requires a restricted domain so that the inverse itself is a function.By limiting the domain of the inverse \(f^{-1}(x) = + \sqrt{(x + 5) / 2}\) to \(x \geq -5\), the range becomes \(y \geq 0\), ensuring we get only one valid, non-negative input output pair that maintains the function characteristic.