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Inverse functions are a type of function that 'undo' the action of the original function. If the function \( f(x) \) takes an input \( x \) and maps it to an output \( y \), then the inverse function, denoted as \( f^{-1}(x) \), takes \( y \) and maps it back to \( x \). Keep in mind a function must be one-to-one for it to have an inverse. If each output \( y \) corresponds to exactly one input \( x \), this function will have an inverse. For example, in our exercise, \( f(x) = x^3 + 5 \) has an inverse because it is one-to-one. To find the inverse, we solve the equation \( y = x^3 + 5 \). We rearrange it to \( x = (y - 5)^{1/3} \), so the inverse function is \( f^{-1}(x) = (x - 5)^{1/3} \).

The Horizontal Line Test is a simple visual way to determine if a function is one-to-one. It involves drawing horizontal lines across the graph of the function. If any horizontal line crosses the graph more than once, the function is not one-to-one. This test is especially useful for visualizing functions quickly. For our function \( f(x) = x^3 + 5 \), we can see that no horizontal line will intersect the graph more than once. Therefore, it passes the Horizontal Line Test. This confirms that the function is one-to-one and thus has an inverse.

Cubic functions are polynomial functions of the form \( f(x) = ax^3 + bx^2 + cx + d \). They are named because the highest power of \( x \) is 3. A key feature of cubic functions is that their graphs have unique shapes and behaviors. For instance, the function \( f(x) = x^3 \) is monotonic, meaning it is always increasing or decreasing. When a constant is added, like in \( f(x) = x^3 + 5 \), the graph shifts up without changing the function's overall shape. This shift does not affect the function's ability to be one-to-one. Hence, \( f(x) = x^3 + 5 \) remains one-to-one and therefore can have an inverse.