Inverse of composite functions


c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.

Verified step by step guidance

1. Recall that a function is one-to-one (injective) if it never assigns the same value to two different domain elements. This means that for functions g and h, if g(a) = g(b), then a must equal b, and similarly for h.
2. Understand that the composition of two one-to-one functions, ƒ(x) = g(h(x)), will also be one-to-one. This is because if g(h(a)) = g(h(b)), then h(a) must equal h(b) (since g is one-to-one), and consequently, a must equal b (since h is also one-to-one).
3. Since ƒ(x) is one-to-one, it has an inverse function, denoted as ƒ⁻¹(y), which will map each output back to a unique input.
4. To find the inverse of the composite function, we can express it as y = g(h(x)). To find the inverse, we will first solve for h(x) in terms of y, and then solve for x in terms of h⁻¹(y).
5. Finally, we conclude that the existence of the inverses g⁻¹ and h⁻¹ allows us to express the inverse of the composite function as ƒ⁻¹(y) = h⁻¹(g⁻¹(y)), confirming that the inverse of ƒ(x) exists.