f(x) = x^3 and g(x) = 3x-7
If the inverses of two functions are both functions, will the inverse of the composite function made by the original functions also be a function?
yes.
(f◦g)(x) = f(g) = g^3 = (3x+7)^3
so,
(f◦g)^-1 (x) = (∛x - 7)/3
Well, if we have two functions, f and g, and both of their inverses are also functions, then it means they are both one-to-one and onto. When we form the composite function f(g(x)), the inverse of that composite function may or may not be a function.
But hey, don't worry! It's not all doom and gloom. We can determine whether the composite function's inverse is a function by checking if the composite function itself is one-to-one. If it is, then the inverse of the composite function will indeed be a function.
So, it's like solving a puzzle - we just need to make sure that all the pieces fit together nicely. Just like a good punchline!
To determine if the inverse of the composite function made by the original functions will be a function, we need to consider the conditions under which the inverse of a function exists.
For the inverse of a function to exist, the original function must be both injective (one-to-one) and surjective (onto). In other words, each input value must correspond to a unique output value, and every output value must have a corresponding input value.
Now, let's analyze the given functions:
f(x) = x^3
g(x) = 3x - 7
To check if f(x) and g(x) are injective, we can examine their slopes.
The slope of f(x) = x^3 is always positive, except at x = 0 where the slope is zero. This means that f(x) is not injective since multiple x values result in the same y value.
The slope of g(x) = 3x - 7 is a constant 3, which means that g(x) is injective, as each x value has a different corresponding y value.
Since f(x) is not injective, its inverse does not exist as a function. However, g(x) is injective, so its inverse does exist.
When we consider the composite function (f∘g)(x), we are essentially applying f(x) to the output of g(x). In other words, (f∘g)(x) = f(g(x)).
To find the inverse of the composite function (f∘g)(x) = f(g(x)), we first need to find the inverse of g(x). Let's call this inverse g^(-1)(x).
Once we find g^(-1)(x), we can substitute it into f(x) to find the inverse of the composite function.
Therefore, the inverse of the composite function will only be a function if the inverse of g(x) exists as a function.