consider the functions f(x)=x^3-2 and g(x)=3 sqrt x+2:
a. find f(g(x))
b. find g(f(x))
c. determine whether the functions f and g are inverse of each other.
I have no clue where to even begin!
f(g) = g^3-2 = (3sqrt(x+2))^3-2 = 27(x+2)sqrt(x+2)-2
g(f) = 3sqrt(f+2) = 3sqrt(x^3) = 3xsqrt(x)
if they are inverses, g(f) = f(g) = x
Obviously they are not inverses
thanks alot
No problem! I'll explain step by step how to find the answers to each question.
a. To find f(g(x)), you need to substitute the expression g(x) into the function f(x). Here's how you can do that:
Step 1: Start with the function f(x) = x^3 - 2.
Step 2: Replace the variable x with the expression for g(x), which is 3√x + 2.
Step 3: Now, we have f(g(x)) = (3√x + 2)^3 - 2.
To simplify further, you need to expand the expression (3√x + 2)^3. You can do this using the binomial expansion or simply by multiplying it out. However, it can be a bit complex to do by hand. If you have access to a graphing calculator or a computer software, you can input the expression (3√x + 2)^3 - 2 and it will give you the simplified form of f(g(x)).
b. To find g(f(x)), you follow a similar process as in Part a.
Step 1: Start with the function g(x) = 3√x + 2.
Step 2: Replace the variable x with the expression for f(x), which is x^3 - 2.
Step 3: Now, we have g(f(x)) = 3√(x^3 - 2) + 2.
Again, you might need a calculator or software to simplify the expression 3√(x^3 - 2) + 2.
c. To determine whether the functions f and g are inverses of each other, you need to check if the composition of f(g(x)) and g(f(x)) simplifies to the identity function, which is x.
So, if f(g(x)) = x and g(f(x)) = x, then f and g are inverses of each other.
To check if f(g(x)) = x, you can simplify the expression obtained in Part a and see if it equals x.
To check if g(f(x)) = x, you can simplify the expression obtained in Part b and see if it equals x.
It's important to simplify the expressions before comparing them with x.
Remember, using a graphing calculator or a software program can make the computations easier and quicker.