Question

For each function, state two ways to restrict the domain so that the inverse is a function a) \(f(x)=x^{2}+4\) b) \(f(x)=2-x^{2}\) b) \(f(x)=2-x^{2}\) c) \(f(x)=(x-3)^{2}\) d)\(f(x)=(x+2)^{2}-4\)

Short answer

Restrict domain of f(x) = x^2 + 4 to x >= 0 or x <= 0. Restrict domain of f(x) = 2 - x^2 to x >= 0 or x <= 0. Restrict domain of f(x) = (x - 3)^2 to x >= 3 or x <= 3. Restrict domain of f(x) = (x + 2)^2 - 4 to x >= -2 or x <= -2.

Step-by-step solution

Understanding the problem

To ensure the inverse of a function is also a function, the original function must be one-to-one (bijective). Functions like quadratics are not naturally one-to-one because they fail the Horizontal Line Test. Therefore, restricting the domain to make the function one-to-one is necessary.

Find the restrictions for a) f(x) = x^2 + 4

For the quadratic function \(f(x) = x^{2} + 4\) to have an inverse, restrict its domain to \(x \geq 0\) or \(x \leq 0\). These restrictions make the function one-to-one.

Find the restrictions for b) f(x) = 2 - x^2

For the quadratic function \(f(x) = 2 - x^{2}\) to have an inverse, restrict its domain to \(x \geq 0\) or \(x \leq 0\). These restrictions make the function one-to-one.

Find the restrictions for c) f(x) = (x - 3)^2

For the quadratic function \(f(x) = (x - 3)^{2}\) to have an inverse, restrict its domain to \(x \geq 3\) or \(x \leq 3\). These restrictions make the function one-to-one.

Find the restrictions for d) f(x) = (x + 2)^2 - 4

For the quadratic function \(f(x) = (x + 2)^{2} - 4\) to have an inverse, restrict its domain to \(x \geq -2\) or \(x \leq -2\). These restrictions make the function one-to-one.

Key concepts

Domain Restriction

When working with functions, especially quadratic functions, we sometimes need to restrict the domain. This is necessary to ensure that the function meets certain criteria, like being one-to-one. By imposing domain restrictions, we control the input values of the function.
Without restricting the domain, many functions, including quadratics, fail to be one-to-one. This means they don't pass the Horizontal Line Test. For instance, the quadratic function \( f(x) = x^2 \) is symmetric around the y-axis.
By restricting the domain to either \( x \geq 0 \) or \( x \leq 0 \), we make the function one-to-one. This makes the inverse of the function also a function.
Another example is the function \( f(x) = (x+2)^2 - 4 \). By restricting the domain to \( x \geq -2 \) or \( x \leq -2 \), we ensure the function behaves in a way that allows for an inverse function.

One-to-One Function

A one-to-one function is vital when we want its inverse to also be a function. This means each input x maps to a unique output y.
For example, linear functions, like \( f(x) = 2x + 3 \), are naturally one-to-one. However, quadratic functions like \( f(x) = x^2 \) are not. They repeat y-values for different x-values.

• To check if a function is one-to-one, use the Horizontal Line Test. Draw horizontal lines across the graph.
• If any line intersects the graph more than once, the function isn't one-to-one.

To make functions like \( f(x) = x^2 \) one-to-one, we restrict their domains. For instance, with \( f(x) = (x-3)^2 \), we can restrict the domain to \( x \geq 3 \) or \( x \leq 3 \) to turn it into a one-to-one function.

Horizontal Line Test

The Horizontal Line Test is a simple way to see if a function is one-to-one. You only need to draw horizontal lines across the graph of the function.
Here's how it works:

• If a horizontal line crosses the graph at more than one point, the function fails the test.
• This means the function is not one-to-one.
• For example, the function \( f(x) = x^2 + 4 \) fails the test without domain restriction since a horizontal line will intersect its parabola twice.

However, restricting the domain can solve this. Limiting the domain to \( x \geq 0 \) or \( x \leq 0 \) ensures that each horizontal line crosses the graph only once.
This converts the original function into a one-to-one function, making its inverse valid.

Quadratic Function

Quadratic functions have the form \( f(x) = ax^2 + bx + c \). They create a parabola on a graph, which is symmetric around its vertex.
However, quadratic functions are not one-to-one by nature. They will fail the Horizontal Line Test because each y-value in the range corresponds to two x-values.
For instance, with \( f(x) = 2 - x^2 \), the graph is a downward-facing parabola centered at the y-axis.
To turn it into a one-to-one function, we need domain restrictions:

• One option is \( x \geq 0 \), limiting the graph to the right side.
• Another option is \( x \leq 0 \), limiting the graph to the left side.

By applying these restrictions, the quadratic function can pass the Horizontal Line Test, allowing us to create an inverse function for it.