Question

PLEASE HELP!!
If f(x)= In x, g(x)=e^6x, and h(x)=x^7
fog (x)=
gof(x)=
foh (x)=

Answer 1

plug them in inside each other
function= input to get output

Answer 2

f(x) = ln (x)
g(x) = e^6x <--- I'm assuming you mean e^(6x) rather than (e^6) * x
h(x) = x^7

fog(x)=f(g(x))
=f(e^6x)
=ln (e^6x)
= 6x

gof(x)=g(f(x))
=g(ln(x))
=e^[6(ln(x))]
=e^(ln(x^6))
=x^6

foh(x)=f(h(x))
=f(x^7)
=ln (x^7)
=7ln(x)

Answer 3

Thank you!

Answer 4

i gets no medal? :/

Answer 5

Yes.... how do I find the domain?

Answer 6

To find the domain of something like "f composed with g, of x", remember that fog(x) means f(g(x)). So the only input values that f will ever see (that is, the only domain values of f), come straight from the output values of g (come straight from the range values of g). From there, you need to consider the domain of f as well.

So, long story short - you have to look at two things. First, find the range of your inside function; that will potentially be the first restriction on your domain. Secondly, find the domain on your outside function.

Let's take the first one for example:
f(x) = ln (x)
g(x) = e^(6x)

The range of g(x) is: (0,infty) so we are automatically affecting the domain of the composition by limiting the values that work as inputs. From there, we look at the domain of ln(x). Here, only positive numbers work, so the domain is (0,infty). Now, conveniently, these both happen to be the same, so the domain for the first one is (0,infty).

*********

Should they ever not be the same, you would use the smaller number set for the domain. As an example, if you had the range for g as [5,infty), and the domain of f as [8,infty), the domain of the composition would be [8,infty) because every number in that set satisfies both the domain of f and the range of g at the same time.



The values between [5,8) don't satisfy the domain of f, so you don't use them.