I need some help with composite functions
where i have to find the f(g(x)) and g(f(x)) on most of them I just couldn't figure out how to multiply....
1) f(x) = ³√x-5 , g(x) = x^3 + 1
So I started on this one and i got confused with the cubed
f(g(x))=³√(x^3+1)-5
then for g(f(x)) = (³√x-5)^3+1
I couldn't figure out how to solve with the cubes please help :) thank you
2) f(x)=√x , g(x) = 2x-3
f(g(x))= √2x-3
and
g(f(x))= 2(√x)-3
Am I done here for finding the f(g(x)) and g(f(x)) ? Or do I have to go a step further?
3) f(x)= x^2/3 , g(x) = x^6
f(g(x))= (x^6)^2/3
and
g(f(x))= (x^2/3)^6
how would I multiply this?
4) f(x)= 3/x^2-1 , g(x) = x+1
f(g(x))= 3/(x+1)^2-1
and
g(f(x))= 3/x^2-1 + 1
1. From your work, I will assume that
f(x) = ³√(x-5)
then f(g(x)) = f(g(x))=³√((x^3+1)-5)
= f(g(x))=³√(x^3 - 4)
test it with some value of x, say x=2
g(2) = 8+1 = 9
f(9) = ³√(9-5) = ³√4
using my answer of f(g(x)) = ³√(x^3 - 4) = ³√(2^3-4) = ³√4
my answer is correct, you just needed the brackets
g(f(x)) = (³√(x-5)^3 + 1 = x-5 + 1 = x - 4
2. Again you will need brackets
f(g(x)) = √(2x-3)
g(f(x)) = yours is correct
3. yes, simplify the exponents by using the exponent rule (x^a)^b = x^(ab)
f(g(x)) = (x^6)^(23) = x^4
g(f(x)) = (x^(2/3))^6 = x^4
4. Did you mean
f(x) = 1/(x^2 - 1) or the way you typed it ?
To solve composite functions, you need to substitute the inner function into the outer function and simplify. Let's go through each of the examples you provided:
1) f(x) = ³√(x-5), g(x) = x^3 + 1
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = ³√((x^3 + 1) - 5)
Simplify the expression inside the cube root:
= ³√(x^3 - 4)
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (³√(x-5))^3 + 1
Simplify:
= (x-5) + 1
= x - 4
2) f(x) = √x, g(x) = 2x-3
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = √(2x-3)
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = 2√x - 3
You're done with both of these, as they are already simplified.
3) f(x) = x^(2/3), g(x) = x^6
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = (x^6)^(2/3)
When you raise a power to another power, you multiply the exponents:
= x^(6 * 2/3)
= x^(12/3)
= x^4
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = (x^(2/3))^6
Multiply the exponents:
= x^(2/3 * 6)
= x^(12/3)
= x^4
In this case, both f(g(x)) and g(f(x)) give you the same result.
4) f(x) = 3/(x^2-1), g(x) = x+1
To find f(g(x)), substitute g(x) into f(x):
f(g(x)) = 3/((x+1)^2-1)
Simplify the expression inside the parenthesis:
= 3/((x+1)*(x+1)-1)
= 3/(x^2 + 2x + 1 - 1)
= 3/(x^2 + 2x)
To find g(f(x)), substitute f(x) into g(x):
g(f(x)) = f(x) + 1 = 3/(x^2-1) + 1
Here, you've successfully found g(f(x)).
Remember, when dealing with exponents, you can simplify by applying exponent rules, such as multiplying exponents together or raising powers to another power. In terms of multiplication, you multiply coefficients or variables as usual.