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?
No one here can answer this. Do you know why?
~~~> (not already chosen by a classmate)
How would anyone here know??
It doesn't need to be a classmate. Rachelle was just looking for an answer bc she didn't know how to do it. All she was looking for was help and anyone smart could've figured it out for her, but apparently, u r not smart enough. So just shut up if u don't know the answer, just don't say anything. (Not that u aren't smart, Rachelle, I am, and I still need help sometimes)
Let's choose two specific functions:
1. f(x) = x^2
2. g(x) = √x
To answer the first question:
If the inverses of two functions, f and g, are both functions, the inverse of the composite function made by the original functions, f(g(x)), will also be a function.
To find the composite function, we substitute g(x) into f(x) as follows:
f(g(x)) = f(√x) = (√x)^2 = x
Now, let's find the inverse of the composite function f(g(x)):
We interchange x and y in the equation and solve for y:
y = x
x = y
Therefore, the inverse of the composite function f(g(x)) is g(f(x)) = g(x^2) = √(x^2) = |x|. Notice that the inverse is still a function.
To answer the second question:
If the inverses of two functions, f and g, are both functions, the inverse of the sum or difference of the original functions, f(x) ± g(x), will also be a function.
To make it clearer, let's consider the sum of the original functions:
h(x) = f(x) + g(x) = x^2 + √x
To find the inverse of the sum, h(x)^-1, we again interchange x and y and solve for y:
y = x^2 + √x
x = y^2 + √y
The equation x = y^2 + √y is not a function because it has two possible outputs for some values of x. Therefore, the inverse of the sum of the original functions is not a function.
For the difference of the original functions, the same conclusion holds true.
To answer the third question:
If the inverses of two functions, f and g, are both functions, the inverse of the product or quotient of the original functions, f(x) * g(x) or f(x) / g(x), may or may not be a function.
Let's consider the product of the original functions:
k(x) = f(x) * g(x) = x^2 * √x = x^(5/2)
To find the inverse of the product, k(x)^-1, we interchange x and y and solve for y:
y = x^(5/2)
x = y^(5/2)
The equation x = y^(5/2) is not a function since it also has two possible outputs for some values of x. Therefore, the inverse of the product of the original functions is not a function.
For the quotient of the original functions, the same conclusion holds true.
In summary, the inverse of the composite function made by two functions will be a function if the inverses of the original functions are both functions, but the inverses of the sum, difference, product, or quotient of the original functions may or may not be functions.
Let's choose the functions f(x) = 2x and g(x) = √x as our example. Both of these functions have inverses.
To answer the first 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?
We can find the composite function by first evaluating g(f(x)). Substituting f(x) = 2x into g(x), we get g(f(x)) = √(2x). We can rewrite this as √(2)x^(1/2), which simplifies to (√2)x^(1/2). Therefore, the composite function g(f(x)) is (√2)x^(1/2).
Now, let's find the inverse of this composite function. The inverse of g(f(x)) would involve finding an x such that (√2)x^(1/2) = y. We can solve this equation by raising both sides to the power of 2 and then dividing by √2. Doing this, we get 2x = (√2)^2y^2, which simplifies to 2x = 2y^2. Dividing both sides by 2, we have x = y^2.
The inverse of the composite function (√2)x^(1/2) is x = y^2. Since we have a clear rule that relates the input y to the output x, we can conclude that the inverse of the composite function is a function. Therefore, if the inverses of two functions are both functions, then the inverse of the composite function made by the original functions will also be a function.
Now, let's move on to the second question: 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?
Let's consider the sum of the original functions, f(x) + g(x). Substituting the functions f(x) = 2x and g(x) = √x into the sum, we have 2x + √x. We don't need to find the inverse of this sum to answer the question.
In this case, we can have a clear answer by using a counterexample. If we let x = 1, then the sum 2x + √x becomes 2(1) + √1 = 2 + 1 = 3. Now, let's find the inverse of 3. There is no input value that we can substitute into the inverse function to obtain 3 as the output. Therefore, the inverse of the sum of the original functions is not a function in general.
Lastly, let's address the third question: 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?
Let's consider the product of the original functions, f(x) * g(x). Substituting the functions f(x) = 2x and g(x) = √x into the product, we have 2x * √x = 2x^(3/2). Similarly to the previous question, we don't need to find the inverse of this product to answer the question.
To answer this question, we can once again use a counterexample. If we let x = 4, then the product 2x * √x becomes 2(4) * √4 = 8 * 2 = 16. Now, let's find the inverse of 16. Again, there is no input value that we can substitute into the inverse function to obtain 16 as the output. Therefore, the inverse of the product of the original functions is not a function in general.
In conclusion, when the inverses of two functions are both functions, the inverse of the composite function made by the original functions will be a function. However, the inverse of the sum or difference of the original functions will not be a function, and the inverse of the product or quotient of the original functions will also not be a function.