Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-3 \sqrt{\frac{4 x-7}{3}} $$

Short answer

The inverse function is \(f^{-1}(x)=3x^2/36+7/4\), and it is indeed a function as it passes the vertical line test.

Step-by-step solution

Write the function in terms of \(y\)

Start by writing the function as \(y=-3 \sqrt{\frac{4 x-7}{3}}\)

Exchange \(x\) and \(y\)

In order to find the inverse function, switch the roles of \(x\) and \(y\). This will yield \(x=-3 \sqrt{\frac{4 y-7}{3}}\)

Solve for \(y\)

We need to get \(y\) by itself. Dividing both sides by -3 gives \(-x/3=\sqrt{\frac{4 y-7}{3}}\). Squaring both sides gives the equation \(x^2/9={4y-7}/3\). Solving this equation for \(y\) will give the inverse function. By expanding it, we get \(3x^2/36=y-7/4\). Adding 7/4 to both sides results in \(y=3x^2/36+7/4=f^{-1}(x)\)

Test if the inverse is a function

A relation is a function if for any value of \(x\), there is at most one value of \(y\). The graph of \(y=3x^2/36+7/4\) is a parabola opening upwards, meaning it will pass a vertical line test. Hence the inverse found is indeed a function.

Key concepts

Function Composition

Function composition is like plugging one function into another. It helps us find how two functions work together. When you have a function, say \(f(x)\), and you plug it into another function \(g(x)\), it's written as \((g \circ f)(x)\), pronounced "g of f of x." This composition provides insight into how changing values in \(f\) affect the output of \(g\).
It's also important when finding inverse functions. For a function and its inverse \(f\) and \(f^{-1}\), respectively, plugging them into each other should give you the starting value: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This property is key to confirming that the functions truly are inverses.

Vertical Line Test

The vertical line test is a simple method to determine if a graph represents a function. Imagine drawing vertical lines over your graph. If any vertical line intersects the graph more than once, it isn't a function.
This test tells us if each input has a single output – a crucial property of functions. When checking inverses, especially with parabolas like in our example \(y=3x^2/36+7/4\), applying the vertical line test confirms if the inverse remains a function.

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle. You adjust equations without changing their inherent value or meaning to isolate variable terms.
In the case of finding inverses, this means rearranging the equation to solve for \(y\). This process includes swapping \(x\) and \(y\), and then isolating \(y\) through operations such as addition, multiplication, division, and more complex functions like square roots.
Each step must preserve equality while gradually revealing the desired form: \(y = f^{-1}(x)\). This ensures that the equation looks different but remains equivalent.

Parabolas

Parabolas are curves that are U-shaped and mirror each other. When graphed, they can open upwards, like a smile, or downward, like a frown. They are often linked to quadratic functions – those with an \(x^2\) term.
Parabolas have a single axis of symmetry and their vertex can either be the highest or the lowest point, depending on the direction they open.
In the solution example, the inverse function results in a parabola that opens upwards. This is significant for confirming that the inverse is a function via the vertical line test. Consequently, although the original function was not a parabola, the inverse was, highlighting the nature and graph behavior of parabolic structures.