Question

In Exercises 21-26, determine whether the function is one-toone on its entire domain and therefore has an inverse function. $$ f(x)=(x+a)^{3}+b $$

Short answer

Yes, the given function \( f(x) = (x+a)^3 + b \) is a one-to-one function on its entire domain, and therefore it has an inverse function.

Step-by-step solution

Analysis of the function

The function given is \( f(x) = (x+a)^3 + b \). The term \( (x+a)^3 \) is a cubic function, which is generally one-to-one. The coefficient of the \( x^3 \) term is positive, which means the function is increasing in its entire domain.

One-to-one Verification by analyzing its derivative

To confirm our analytical observation, we can compute the derivative of this function. The derivative of \( f(x) = (x+a)^3 + b \) is \( f'(x) = 3(x+a)^2 \). As the function \( (x+a)^2 \) is always non-negative for any real number \( x \), and multiplicative factor 3 is always greater than zero, the derivative of this function is always non-negative (which means the function is not decreasing). But we already know that \( (x+a)^3 \) is increasing over the entire real line, so derivative cannot be zero TOO. Hence confirmed that the function is strictly increasing over the entire real line, which confirms it's one-to-one.

Conclusion

Since we have determined that \( f(x) = (x+a)^3 + b \) is strictly increasing everywhere in its domain due to \( f'(x) > 0 \) for all \( x \), it must be a one-to-one function. Therefore, it has an inverse function, which can be obtained by swapping \( x \) and \( y \), and then solving the equation for \( y \).