Day 10 HW Putting it all together We've now seen linear, exponential, and power functions. We've also seen their inverses (when and where they exist). It is a fact that if f and g are invertible functions, and if the composition fog is defined, then fog is also invertible, and its inverse is (fog)-1=f¹og¹. For example: f(x)=x+2 has inverse f(x)=x-2, and g(x)=x has inverse g(x)=√√x, so (fog)(x)=3+2 has inverse (fog)(x)=g¹oƒ˜¹=√√x−2. (You should check that this works!) The fact that the inverses get composed in the opposite order is often explain like this: In the morning, you first put on your socks, then you put on your shoes; at night, you must first take off your shoes before you can take off your socks. For each of the following functions, decompose it has a composition of two functions, and use this to compute its inverse. 1. f(x)=√x-1 2. f(x) = (2x+3)² Day 10 HW Putting it all together We've now seen linear, exponential, and power functions. We've also seen their inverses (when and where they exist). It is a fact that if ƒ and g are invertible functions, and if the composition f@g is defined, then f@g is also invertible, and its inverse is (f@g) - 1 = f¹¹@g¹. For example: f(x) = x + 2 has inverse f¹(x) = x-2, and 3 3 9(x) = x³ has inverse g(x) = ³√x, so (f@g)(x) = x³ + 2 has inverse (f@g)(x) = g¨¹@ƒ¨¹ = == = ³√x - 2. (You should check that this works!) The fact that the inverses get composed in the opposite order is often explain like this: In the morning, you first put on your socks, then you put on your shoes; at night, you must first take off your shoes before you can take off your socks. For each of the following functions, decompose it has a composition of two functions, and use this to compute its inverse. f(x)=x-1 = √x-1 f(x) = (2x+3)² f(x) = In(x^2) f(x) = x^7-4 f(x) = 2x^x+1