Verify that the functions f and g are inverses of each other by showing f(g(x)) = x and g(f(x)) = x
f(x) = x^3 + 5
g(x) = 3sqrtx-5 ( 3 is inside check mark on the sqrt.
I am sooo totally lost on these!
g(x) takes input of "x" and gives output of "cuberoot(x - 5)" (that's what the little 3 means outside the root sign).
Imagine feeding that into the f function.
Note that f(anything) = (anything)^3 + 5.
So suppose we put in what we got from doing g(x). What we got was "cubert(x - 5)". So let's put that into the f function.
f(cubert(x-5)) = (cubert(x-5))^3+5
But (cubert(x-5))^3 is just (x-5).
And so (x-5)+5 is just x.
This is how you can show that f(g(x)) = x.
The nesting of parentheses helps you see the order in which the functions are applied. You read from the innermost out. So first function g gets applied to x and gives an expression that we can call g(x). Then the function f works on that g(x) as input and gives an output. Performing one function on the result of an earlier function is called composing functions, or composition of functions.
So you've seen how f(g(x)) = x.
Can you prove that g(f(x)) = x?
Start with x and apply function f to it. Then take the result and apply function g to that, using f(x) wherever "x" appears in the rule for g(x).
Not coming out right!!!!!!
No worries! Verifying whether two functions are inverses of each other involves checking if "applying one function and then the other" yields the original input. In other words, we need to show that f(g(x)) = x and g(f(x)) = x for all x in their domain.
Let's start with f(g(x)) = x:
Step 1: Substitute g(x) into f(x).
Replace x in f(x) with g(x) to get:
f(g(x)) = (g(x))^3 + 5
Step 2: Substitute the expression for g(x).
Replace g(x) with its expression:
f(g(x)) = (3√x - 5)^3 + 5
Step 3: Simplify the expression.
Expand and simplify the expression using appropriate rules:
f(g(x)) = (27x - 135√x + 225 - 15√x + 75x^(1/2) - 125x^(3/2) + 5
Step 4: Combine like terms.
Combine like terms to simplify the expression further:
f(g(x)) = -125x^(3/2) + 75x^(1/2) + 27x - 150√x + 230
Step 5: Simplify the expression further if possible.
If there are any simplifications that can be made further, do so. However, based on the given function f(x) = x^3 + 5, it seems unlikely that this expression can be simplified further.
Step 6: Verify f(g(x)) = x.
Now, we need to verify if f(g(x)) is equal to x:
f(g(x)) = -125x^(3/2) + 75x^(1/2) + 27x - 150√x + 230
We cannot immediately verify f(g(x)) = x by inspection because it seems complicated. However, we can simplify this process by selecting a specific value of x and substituting it into both f(g(x)) and x to check if they are equal.
For example, let's choose x = 2:
f(g(2)) = -125(2)^(3/2) + 75(2)^(1/2) + 27(2) - 150√2 + 230
= -125√2 + 150 + 54 - 150√2 + 230
= 404 - 275√2
Since f(g(2)) results in a numerical expression 404 - 275√2, it is difficult to compare it directly to x = 2. Hence, we cannot conclude whether f(g(x)) = x at this point.
Similarly, we need to check g(f(x)) = x:
Step 1: Substitute f(x) into g(x).
Replace x in g(x) with f(x) to get:
g(f(x)) = 3√((x^3 + 5)) - 5
Step 2: Simplify the expression, if possible.
Based on the given function g(x) = 3√x - 5, we cannot simplify this expression further.
Step 3: Verify g(f(x)) = x.
Now, we need to verify if g(f(x)) is equal to x:
g(f(x)) = 3√((x^3 + 5)) - 5
Again, we can simplify the process by selecting a specific value of x and substituting it into both g(f(x)) and x to check if they are equal. Let's choose x = 2:
g(f(2)) = 3√((2^3 + 5)) - 5
= 3√(8 + 5) - 5
= 3√(13) - 5
Similarly to the previous case, the expression 3√(13) - 5 is not directly comparable to x = 2, so we cannot determine if g(f(x)) = x at this point.
In conclusion, due to the complexity of the expressions resulting from both f(g(x)) and g(f(x)), we cannot definitively determine whether f(x) = x^3 + 5 and g(x) = 3√x - 5 are inverses of each other. It is possible that there has been a mistake in the given functions or further simplifications are required.