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:

Question

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?

Answer 1

the inverse of f(x)=x^3 is f(x)=3sqrt x;this is a function.the inverse of g(x)=3x-7 is y=x/3 + 7/3;this is a function.what is the composite function what does that mean?

Answer 2

so, did you pick two functions?

what do you need to make sure the inverses are also functions?
How about
f(x) = x^3 and g(x) = 3x-7
now form the composites and sums, etc.
Are they functions?

Answer 3

thanks steven the only problem is i am having a hard time with calculus i dont really understand any of this ..

Answer 4

Be sure to make good use of https://www.khanacademy.org and https://wolframalpha.com

Khanacademy is especially good at providing explanations. Be sure to make an account for your self (it's free) so you can track your progress, and if you take any of their quizzes, you'll get feedback.

Answer 5

since

f^-1(x) = ∛x
g^-1(x) = (x+7)/3

(f^-1 ◦ g^-1)(x) = ∛(g^-1) = ∛((x+7)/3) which is a function.
(g^-1 ◦ f^-1)(x) = (f^-1 + 7)/3 = (∛x + 7)/3 which is a function

as long as f and g are strictly increasing or decreasing, they are 1:1, so they have inverse functions.

Answer 6

Let's choose two specific functions: f(x) = sqrt(x) and g(x) = x^2. We will use these functions to answer the question "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?"

To find the inverse of a function, we can swap the x and y variables and solve for y.

For f(x), we have y = sqrt(x). To find its inverse, we swap x and y and solve for y:

x = sqrt(y)
Squaring both sides, we have x^2 = y
So, the inverse of f(x) = sqrt(x) is f^(-1)(x) = x^2.

For g(x), we have y = x^2. To find its inverse, we swap x and y and solve for y:

x = y^2
Taking the square root of both sides, we have sqrt(x) = y
So, the inverse of g(x) = x^2 is g^(-1)(x) = sqrt(x).

Now, let's find the composite function of f(g(x)):

f(g(x)) = f(x^2) = sqrt(x^2) = |x|

The inverse of |x| is |x|. Therefore, the inverse of the composite function f(g(x)) is f^(-1)(x) = |x|.

Since the inverse of the composite function is a function, we can conclude that if the inverses of two functions are both functions, the inverse of the composite function made by the original functions will also be a function.