1.) If h(x)= (fog)(x) and h(x) = 3 over the square root of x+3 , find g(x) if f(x)= 3 over the square root of x+2.
my answer is x+1. Am I right?
2.)If f(x)=3x,g(x)=x-3,and h(x)= the square root of x, find (fogox)(X). I really didn't get this will someone please show me how to do it?
The first one is correct but I am still not sure about the second one.
This I what I did to solve for it:
3(x-3)
3(he square root of 25-3
3(5-3)
15-9=6
Is it correct?
f(x) = 3/√(x+2)
f(g) = 3/√(g+2) = 3/√(x+3)
so, you are correct; g(x) = x+1
for the second one, you're not solving any equation; you're just expressing a composite function as a single one, just as in #1.
f(x) = 3x
g(x) = x-3
h(x) = √x
(f◦g◦h)(x) = f(g(h(x)))
f(g(h)) = f(h-3) = 3(h-3) = 3(√x-3)
or,
f(g(h)) = 3(g(h)) = 3(g(√x)) = 3(√x-3)
To find g(x) in the first question, we can start by substituting f(x) into the equation h(x) = (fog)(x):
h(x) = 3 / sqrt(x + 3)
Since h(x) = (fog)(x), this means h(x) is equal to f(g(x)). Rewrite h(x) using g(x):
h(x) = f(g(x)) = 3 / sqrt(g(x) + 2)
Now, compare this to the given equation for h(x):
3 / sqrt(x + 3) = 3 / sqrt(g(x) + 2)
To find g(x), we need to equate the expressions inside the square root:
x + 3 = g(x) + 2
Rearranging the equation, we get:
g(x) = x + 1
Therefore, the correct answer for g(x) is indeed x + 1.
For the second question, to find (fogox)(x), we need to substitute g(x) and f(x) into the equation:
(fogox)(x) = f(g(h(x))) = f(g(sqrt(x)))
First, find g(sqrt(x)) by substituting sqrt(x) into the equation for g(x):
g(sqrt(x)) = sqrt(x) - 3
Next, substitute the expression for g(sqrt(x)) into the equation for f(x):
f(g(sqrt(x))) = 3(g(sqrt(x)) + 2) / sqrt(g(sqrt(x)) + 2)
Substitute the value of g(sqrt(x)):
f(g(sqrt(x))) = 3(sqrt(x) - 3 + 2) / sqrt(sqrt(x) - 3 + 2)
Simplify:
(fogox)(x) = [3(sqrt(x) - 1)] / sqrt(sqrt(x) - 1)
So, (fogox)(x) = [3(sqrt(x) - 1)] / sqrt(sqrt(x) - 1)
1) To find g(x), we need to solve for g(x) from the equation h(x) = (f∘g)(x).
Given that h(x) = 3 / √(x + 3) and f(x) = 3 / √(x + 2), we can substitute h(x) and f(x) into the equation for (f∘g)(x):
h(x) = (f∘g)(x)
3 / √(x + 3) = 3 / √(g(x) + 2)
To solve for g(x), we can start by isolating √(g(x) + 2) on one side of the equation:
3 / √(x + 3) = 3 / √(g(x) + 2)
√(g(x) + 2) = √(x + 3)
Next, we square both sides of the equation to eliminate the square root:
(g(x) + 2) = (x + 3)
We can then isolate g(x) by subtracting 2 from both sides of the equation:
g(x) = x + 3 - 2
g(x) = x + 1
So, the value of g(x) is x + 1.
Therefore, your answer of g(x) = x + 1 is correct.
2) To find (f∘g∘h)(x), we need to evaluate the composite function by performing the operations in the correct order.
Given that f(x) = 3x, g(x) = x - 3, and h(x) = √x:
(f∘g∘h)(x) = f(g(h(x)))
First, we substitute h(x) into g(x):
g(h(x)) = h(x) - 3 = √x - 3
Next, we substitute g(h(x)) into f(x):
f(g(h(x))) = f(√x - 3) = 3(√x - 3)
We can simplify this expression by distributing the constant 3 to both terms inside the parentheses:
f(g(h(x))) = 3√x - 9
So, (f∘g∘h)(x) = 3√x - 9.