Choose any two specific functions (not already chosen by a classmate) that have inverses. Use your chosen functions to answer any one of the following questions.
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?
If the inverses of two functions are both functions, will the inverse of the sum or difference of the original functions also be a function?
If the inverses of two functions are both functions, will the inverse of the product or quotient of the original functions also be a function?
so, which functions have you chosen?
How do we know that no one else has already chosen them?
I haven't chosen anything, I don't even know what's going on in this class, I have a week left to complete this course, I just started. Any function is fine, no one has responded in the course I'm in.
Let's choose two specific functions, f(x) = 2x and g(x) = x/3, both of which have inverses.
To answer the first question, we need to find the inverse of the composite function made by the original functions. The composite function is formed by taking the output of one function as the input of the other.
Let's denote the composite function as h(x) = f(g(x)). To find the inverse of h(x), we need to find a function h^(-1) such that h(h^(-1)(x)) = x for all x in the domain of h(x).
Now, let's find the inverse of f(x) = 2x. To do this, we swap the roles of x and f(x) and solve for x:
x = 2f^(-1)(x)
x/2 = f^(-1)(x)
So, the inverse of f(x) is f^(-1)(x) = x/2.
Next, let's find the inverse of g(x) = x/3:
x = g^(-1)(x)/3
3x = g^(-1)(x)
g^(-1)(x) = 3x
Now, let's find the composite function h(x) = f(g(x)):
h(x) = f(g(x)) = f(x/3) = 2(x/3) = 2x/3
To find the inverse of h(x), we swap the roles of x and h(x) and solve for x:
x = 2h^(-1)(x)/3
3x = 2h^(-1)(x)
h^(-1)(x) = 3x/2
As we can see, the inverse of the composite function h(x) = f(g(x)) is h^(-1)(x) = 3x/2, which is a function.
Therefore, if the inverses of two functions are both functions, the inverse of the composite function made by the original functions is also a function.
Now, let's answer the second question. We need to determine if the inverse of the sum or difference of the original functions is also a function.
Let's consider the sum of the original functions, f(x) + g(x):
(f + g)(x) = f(x) + g(x) = 2x + x/3
To find the inverse of (f + g)(x), we swap the roles of x and (f + g)(x):
x = 2(f + g)^(-1)(x) + (f + g)^(-1)(x)/3
Now, if we substitute (f + g)^(-1)(x) = a, we get:
x = 2a + a/3
3x = 6a + a
3x = 7a
a = 3x/7
As we can see, the inverse of the sum of the original functions, (f + g)^(-1)(x) = 3x/7, is also a function.
Therefore, if the inverses of two functions are both functions, the inverse of the sum of the original functions is also a function.
Finally, let's answer the third question. We need to determine if the inverse of the product or quotient of the original functions is also a function.
Let's consider the product of the original functions, f(x) * g(x):
(f * g)(x) = f(x) * g(x) = 2x * x/3 = 2x^2/3
To find the inverse of (f * g)(x), we swap the roles of x and (f * g)(x):
x = (f * g)^(-1)(x) * 2 + 3/(f * g)^(-1)(x)
Now, if we substitute (f * g)^(-1)(x) = b, we get:
x = 2b + 3/b
Unfortunately, this equation is not a simple linear equation that can be solved algebraically. Therefore, we cannot determine if the inverse of the product of the original functions is a function or not.
Hence, if the inverses of two functions are both functions, we cannot conclusively determine if the inverse of the product or quotient of the original functions is also a function.