Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=2 x^3-5 $$

Short answer

The inverse of function \(f(x)=2 x^3-5\) is indeed a function based on the horizontal line test and the inverse of function \(f(x)\) is \[ f^{-1}(x)= \sqrt[3]{\frac{x+5}{2}} \]

Step-by-step solution

Determine if the original function is a function

In order to determine that the inverse of a function will also be a function, the original function should pass the Horizontal Line Test. This means, any horizontal line drawn through the function will touch at more than one point. However, the function \(f(x)=2 x^3-5\) is a cubic function and will pass the horizontal line test. Hence, its inverse will be a function as well.

Find the inverse of the function

To find the inverse of a function, we need to switch \(x\) and \(y\), and then solve for \(y\). Our given function is \(y = 2x^3-5\); switching \(x\) and \(y\) will give us \(x = 2y^3-5\). Solving for \(y\) we find the inverse function \[ f^{-1}(x)= \sqrt[3]{\frac{x+5}{2}} \]

Confirm

We use the confirmation that if \(f(g(x)) = x\) for every \(x\) in the domain of \(g\), and \(g(f(x)) = x\) for every \(x\) in the domain of \(f\), then \(f\) and \(g\) are inverse functions. Here \(g(x)\) is the found inverse, which we already know is also a function.

Key concepts

Horizontal Line Test

The Horizontal Line Test is a simple method to help verify if a function has an inverse that is also a function. Imagine drawing horizontal lines across the graph of your function. If each horizontal line crosses the graph at most once, then the function passes the test. This means the function is one-to-one and has an inverse that is also a function. In the case of cubic functions like the one given by \(f(x) = 2x^3 - 5\), the curve generally moves continuously through positive and negative infinity, allowing horizontal lines to intersect only once at any height. This confirms that a cubic function will pass the Horizontal Line Test, as it maintains a one-to-one relationship between inputs and outputs.

Cubic Functions

Cubic functions are polynomial functions with an equation of the form \(f(x) = ax^3 + bx^2 + cx + d\). In such functions, the highest degree of the variable \(x\) is 3, which gives the polynomial its characteristic 'cubic' identity. These functions usually display a characteristic "S" shape curve, which might include one peak and one trough, but this is dependent on coefficients' values. In simpler forms like \(2x^3 - 5\), the graph typically doesn't loop back on itself, which ensures it is one-to-one. The important thing to note for cubic functions is that if they are of the form \(ax^3 + d\) (without \(bx^2\) or \(cx\) terms), they are strictly increasing or decreasing and will definitely have inverses that are functions.

Solving Equations

Finding the inverse of a function involves solving equations in a specific way. First, you start with your original function, say \(f(x) = 2x^3 - 5\). You replace \(f(x)\) with \(y\) to get \(y = 2x^3 - 5\), then switch \(x\) and \(y\), arriving at \(x = 2y^3 - 5\). The next step is to isolate \(y\). Here, begin by adding 5 to both sides: \[x + 5 = 2y^3\]Then, divide everything by 2:\[\frac{x + 5}{2} = y^3\]Finally, take the cube root of both sides to solve for \(y\):\[y = \sqrt[3]{\frac{x + 5}{2}}\]Thus, the inverse function is \(f^{-1}(x) = \sqrt[3]{\frac{x + 5}{2}}\). Each of these steps reflects the principles of algebra, involving reversals of addition, multiplication, and applying roots, maintaining the balance throughout the equation to correctly solve for \(y\).