91Ó°ÊÓ

When working with functions and their inverses, understanding domain and range is essential. The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

For the given function \( f(x) = x^3 - 4 \), we can see that it’s a cubic function. Generally, cubic functions have a domain of all real numbers, denoted by \( (-∞, ∞) \). This means that no matter what real number you input into the function, you will always get a valid output.

Similarly, the range of a cubic function is also all real numbers, because the output can span from negative to positive infinity as x varies through all real numbers. Thus, the range of \( f(x) \) is also \( (-∞, ∞) \).

Now, when finding the inverse function, \( f^{-1}(x) \), the domain and range of the inverse function are swapped from the original function. So, the domain of \( f^{-1}(x) = (x + 4)^{1/3} \) is \( (-∞, ∞) \) and its range is also \( (-∞, ∞) \).

Graphing functions helps in visualizing their behavior and understanding the relationship between the function and its inverse. To graph \( f(x) = x^3 - 4 \) and its inverse \( f^{-1}(x) = (x + 4)^{1/3} \) on the same axes, follow these steps:

1. **Plot Points for \( f(x) \)**: Choose several x-values and compute the corresponding y-values. For example:
- For \( x = -2 \), \( f(-2) = (-2)^3 - 4 = -12 \)
- For \( x = 0 \), \( f(0) = 0^3 - 4 = -4 \)
- For \( x = 2 \), \( f(2) = 2^3 - 4 = 4 \)

2. **Connect the Points Smoothly**: Since \( f(x) \) is a cubic function, the graph will be a smooth curve passing through the points.

3. **Plot Points for \( f^{-1}(x) \)**: Again, choose several x-values and compute the corresponding y-values for the inverse function. For example:
- For \( x = -3 \), \( f^{-1}(-3) = (-3 + 4)^{1/3} = 1^{1/3} = 1 \)
- For \( x = 0 \), \( f^{-1}(0) = (0 + 4)^{1/3} = 1.59 \)
- For \( x = 4 \), \( f^{-1}(4) = (4 + 4)^{1/3}= 2 \)

4. **Reflect Across Line \( y = x \)**: Note that the graph of \( f^{-1}(x) \) will be a reflection of the graph of \( f(x) \) across the line \( y = x \).

The visualization helps understand that for each corresponding value in \( f(x) \), there’s a mirrored value in \( f^{-1}(x) \).

A composite function is formed when one function is applied to the results of another function. When we talk about composite functions involving a function and its inverse, we commonly examine \( f(f^{-1}(x)) \) and \( f^{-1}(f(a)) \).

Ensuring composite functions return to the original value is a crucial property of true inverse functions. Let's look deeper:

1. Finding \( f(f^{-1}(5)) \):
- First, find \( f^{-1}(5) \):
\( f^{-1}(5) = (5 + 4)^{1/3} = 9^{1/3} = 2.08 \)
- Next, substitute this result back into the function \( f \):
\( f(2.08) = (2.08)^3 - 4 = 9 - 4 = 5 \)
This confirms that \( f(f^{-1}(5)) = 5 \).

2. Finding \( f^{-1}(f(a)) \):
- First, substitute \( a \) into the function \( f \):
\( f(a) = a^3 - 4 \)
- Next, substitute this result into the inverse function \( f^{-1} \):
\( f^{-1}(a^3 - 4) = ((a^3 - 4) + 4)^{1/3} \)
Simplify:
\( (a^3)^{1/3} = a \)
So, \( f^{-1}(f(a)) = a \).

This shows that applying the function and its inverse in succession brings you back to your original starting value. This inherent property is a great way to check if two functions are truly inverses of each other.