Question

The function \(f\) is one-to-one. (a) Find its inverse function \(f^{-1}\) and check your answer. (b) Find the domain and the range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{2 x-3}{x+4} $$

Short answer

The inverse function is \( f^{-1}(x) = \frac{-3 - 4x}{x - 2} \). The domain of \( f(x) \) is all real numbers except \( x = -4 \), and the range is all real numbers except \( y = 2 \). For \( f^{-1}(x) \), the domain is all real numbers except \( x = 2 \), and the range is all real numbers except \( y = -4 \).

Step-by-step solution

Express the Function as y

Let \( f(x) = y \). Therefore, \( y = \frac{2x - 3}{x + 4} \).

Swap x and y to Find the Inverse

Switch \( x \) and \( y \) to get \( x = \frac{2y - 3}{y + 4} \).

Solve for y

Multiply both sides by \( y + 4 \): \( x(y + 4) = 2y - 3 \). This expands to \( xy + 4x = 2y - 3 \). Rearrange to group the \( y \) terms on one side: \( xy - 2y = -3 - 4x \), which simplifies to \( y(x - 2) = -3 - 4x \). Now, solve for \( y \): \( y = \frac{-3 - 4x}{x - 2} \). Thus, the inverse function is \( f^{-1}(x) = \frac{-3 - 4x}{x - 2} \).

Verify the Inverse Function

To verify, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Compute \( f\left(\frac{-3 - 4x}{x - 2}\right) \): \( f\left(\frac{-3 - 4x}{x - 2}\right) = \frac{2\left(\frac{-3 - 4x}{x - 2}\right) - 3}{\left(\frac{-3 - 4x}{x - 2}\right) + 4} \). Simplifying this, we get \( x \). Similarly, for \( f^{-1}(f(x)) \), substitute \( x \) into \( f^{-1} \) and verify it equals \( x \).

Find the Domain and Range of f

The domain of \( f(x) \) is all real numbers except \( x = -4 \) (as it makes the denominator zero). The range is all real numbers except \( y = 2 \) (as there is a horizontal asymptote at \( y = 2 \)).

Find the Domain and Range of f^{-1}

The domain of \( f^{-1}(x) \) is all real numbers except \( x = 2 \), corresponding to the range of \( f(x) \). The range of \( f^{-1}(x) \), corresponding to the domain of \( f(x) \), is all real numbers except \( y = -4 \).

Key concepts

One-to-one functions

A function is termed one-to-one if every unique input yields a unique output. In simpler terms, each value of the domain maps to a different value in the range. There are no repeated y-values for different x-values.
You can test if a function is one-to-one using the Horizontal Line Test. If any horizontal line intersects the graph of the function in more than one place, the function is not one-to-one.
In the exercise, the function given is one-to-one:
\[ f(x)=\frac{2 x-3}{x+4} \] This ensures that it has an inverse function.

Domain and range

The domain and range of functions deal with input and output values, respectively. The domain of a function is the complete set of possible values of the independent variable (x-values). The range is the set of possible output values (y-values).
Let's analyze the domain and range for the general function and its inverse:

• The domain of \[ f(x)\ (\frac{2 x-3}{x+4}) \] excludes any value of x that would make the denominator zero. Hence, x cannot be -4.
• The range of f(x) is found by observing the behavior of the function as x approaches the value that makes the denominator zero. Here, y approaches 2, so y cannot be 2.
• For the inverse function \[ f^{-1}(x)= \frac{-3 - 4x}{x - 2} \] ,
• the domain now excludes the value that makes the denominator zero. Hence, x cannot be 2.
• The range of the inverse function will be similar to the original function's domain. Hence, y cannot be -4.

Verifying inverse functions

Verifying an inverse function requires demonstrating that applying one function and then the other returns the original value.
For a function f and its supposed inverse g, we need to show:
\[ f(g(x))=x\ and\ g(f(x))=x \]
Let's verify this for the given functions:
Apply the inverse function to the original \[ f\left(\frac{-3 - 4x}{x - 2}\right) \]:

• Substituting back into f: \[ \frac{2\left(\frac{-3 - 4x}{x - 2}\right) - 3}{\left(\frac{-3 - 4x}{x - 2}\right) + 4} = x \] gives the original input x.
• Now, apply the original function to the inverse: \[ g\left(f(x)\right) = \frac{-3 - 4\left(\frac{2x - 3}{x + 4}\right)}{\left(\frac{2x - 3}{x + 4}\right) - 2} \] also simplifies back to x.

By verifying these compositions, we ensure that the functions are indeed inverses of each other.