Question

In Exercises 23–28, find the inverse of the function. Then graph the function and its inverse. $$ f(x)=4 x^2, x \leq 0 $$

Short answer

The inverse of the given function is \(f^{-1}(x) = -\sqrt{x/4}\), and the domain of this inverse function is \(x \leq 0\).

Step-by-step solution

Swap x and y

Instead of \(f(x) = 4x^2\), write it as \(y = 4x^2\). To find the inverse, swap x and y, which gives \(x = 4y^2\).

Solve for y

To find \(y\), we can divide both sides by 4 resulting in \(y^2 = x/4\), and then take the square root. But as \(x \leq 0\) in the original function, the inverse function, which is the square root, should take a negative sign. So the inverse of the function is \(y = -\sqrt{x/4}\) or \(f^{-1}(x) = -\sqrt{x/4}\).

Find the domain

The domain of the inverse function is the range of the original function. Since the original function is defined for \(x \leq 0\), therefore for the inverse function \(f^{-1}(x)\), x is also less than or equal to 0.

Key concepts

Graphing Functions

Graphing functions is an essential skill when it comes to understanding their behavior and characteristics. In the context of finding inverse functions, graphing becomes a helpful tool because it allows us to visualize both the function and its inverse.

• Start by plotting the original function, if possible, on a coordinate plane. For the given function, this means plotting points for different values of \(x\) that are \(x \leq 0\).
• Since the function is \(f(x) = 4x^2\), it is a parabola opening upwards. However, because \(x \leq 0\), only the left side of the parabola will be plotted.
• To graph the inverse function, notice that each point \((a, b)\) on the original function corresponds to the point \((b, a)\) on the inverse function.
• Swap the coordinates for each point on the function to get the points on the inverse. Graph the inverse to see its relationship with the original.

Both functions, the original and its inverse, should mirror each other across the line \(y = x\). This is a key property of inverse functions that graphing beautifully demonstrates.

Quadratic Functions

Quadratic functions, such as \(f(x) = 4x^2\), are among the most common types of polynomial functions. They are characterized by their parabolic shape. The graph of a quadratic function is a curve where its general form is \(ax^2 + bx + c\).

• In our case, the quadratic function \(f(x) = 4x^2\) has \(a = 4\), and \(b = 0\), \(c = 0\), making it a simple parabola that opens upwards.
• Because \(x \leq 0\), we only consider the left-hand side of this parabola for our specific function. This constraint makes the function and graph non-bijective but allows for finding its inverse within these limits.
• Notice the importance of restricting the domain to ensure that the function is one-to-one. Only a one-to-one function has an inverse, ensuring each input corresponds to a unique output.

Square Roots

Square roots are an essential concept in algebra, especially when dealing with inverse functions of quadratic equations. When finding the inverse of our quadratic function, we naturally encounter a square root.

• We rewrote \(x = 4y^2\) to \(y^2 = \frac{x}{4}\) and took the square root to solve for \(y\).
• In our case, we obtained \(y = -\sqrt{x/4}\) due to the negative domain constraint \(x \leq 0\) from the original function.
• Understanding how to work with square roots is critical, especially when interpreting how the inverse function \(f^{-1}(x) = -\sqrt{x/4}\) behaves. The negative sign in the square root reflects the restriction of the original function's domain on its inverse.

Square roots can take on either the positive or negative value, but domain restrictions help clarify which value is applicable and ensure the function remains valid within the proper mathematical context.