1. **Problem statement:** We want to find the condition under which a composite function has an inverse. 2. **Recall definitions:** - A composite function is defined as $ (f \circ g)(x) = f(g(x)) $. - A function has an inverse if it is bijective (both injective and surjective). 3. **Key formula and rule:** - For $ f \circ g $ to have an inverse, it must be bijective. - This means $ f \circ g $ must be one-to-one (injective) and onto (surjective). 4. **Injectivity condition:** - If $ f \circ g $ is injective, then for any $ x_1, x_2 $, if $ (f \circ g)(x_1) = (f \circ g)(x_2) $, then $ x_1 = x_2 $. - This implies $ g(x_1) = g(x_2) $ must imply $ x_1 = x_2 $ (so $ g $ is injective), and also $ f $ must be injective on the image of $ g $. 5. **Surjectivity condition:** - For $ f \circ g $ to be surjective onto the codomain, $ f $ must be surjective onto the codomain, and $ g $ must be surjective onto the domain of $ f $. 6. **Summary:** - The composite function $ f \circ g $ has an inverse if and only if $ g $ is injective, $ f $ is injective on the image of $ g $, and $ f $ is surjective onto the codomain. 7. **In simpler terms:** - Both $ f $ and $ g $ must be invertible functions, and the range of $ g $ must match the domain of $ f $. **Final answer:** $$\text{The composite function } f \circ g \text{ has an inverse if and only if } g \text{ is injective and } f \text{ is bijective on the image of } g.$$