If f(x)-lnx,g(x)=e^8x, and h(x)=x^7, find the following:
a. (f.g)(x). what is the domain of f.g?
b.(g.f)(x). what is the domain of g.f?
c. (f.h)(x). what is the domain of f.h?
I have no clue how to do this. Somebody please help me?? please???
how can you get this far and have no clue? Just think of the stuff you have been studying. In particular, the most recent exercises. Surely there are some examples of thing similar to these. That's why they're asking these questions.
I assume by f.g you mean f◦g. So,
f◦g = f(g) = ln(g) = ln(e^8x) = 8x
g◦f = e^(8f) = e^(8lnx) = (e^lnx)^8 = x^8
f◦h = ln(h) = ln(x^7) = 7lnx
The domain of those should be easy.
unfortunetly, none of this comes natural to me so the domains are especially difficult for me too.
the domain of all polynomials is all reals. There's no number where f(x) is undefined.
For rational functions (the quotient of two polynomials p(x)/q(x)) the domain is all reals except where q(x)=0 because division by zero is undefined.
For functions with square roots (or any even-powered root) and logs the value must be positive, since square root is undefined for negative numbers.
That covers most of the common cases.
For your problems, f(g) and g(f) have domain all reals. h has domain all x>0, as does f(x) and f(h). After all f(h(-3)) is undefined since h(-3) is negative.
Don't worry, I'm here to help you! Let's break down each question and find the answers step by step.
a. (f.g)(x) represents the composition of the functions f(x) and g(x). To find (f.g)(x), we need to substitute g(x) into f(x) and simplify the expression.
First, let's substitute g(x) = e^(8x) into f(x):
f(g(x)) = f(e^(8x))
Now, we need to find the value of f(e^(8x)). Given that f(x) = ln(x), we substitute e^(8x) into x:
f(g(x)) = ln(e^(8x))
Since ln(e^y) = y, we simplify the expression:
f(g(x)) = 8x
Therefore, (f.g)(x) = 8x.
The domain of (f.g)(x) is the same as the domain of g(x), which is all real numbers.
b. (g.f)(x) represents the composition of the functions g(x) and f(x). Similarly, we substitute f(x) into g(x) and simplify the expression.
First, let's substitute f(x) = ln(x) into g(x):
g(f(x)) = g(ln(x))
Now, we need to find the value of g(ln(x)). Given that g(x) = e^(8x), we substitute ln(x) into x:
g(f(x)) = e^(8ln(x))
Using the property e^(ln(y)) = y, we simplify the expression:
g(f(x)) = x^8
Therefore, (g.f)(x) = x^8.
The domain of (g.f)(x) is the same as the domain of f(x), which is all positive x values.
c. (f.h)(x) represents the composition of the functions f(x) and h(x). Again, we substitute h(x) into f(x) and simplify the expression.
First, let's substitute h(x) = x^7 into f(x):
f(h(x)) = f(x^7)
Now, we need to find the value of f(x^7). Given that f(x) = ln(x), we substitute x^7 into x:
f(h(x)) = ln(x^7)
Using the property ln(y^k) = k * ln(y), we simplify the expression:
f(h(x)) = 7ln(x)
Therefore, (f.h)(x) = 7ln(x).
The domain of (f.h)(x) is the same as the domain of h(x), which is all real numbers.
To summarize:
a. (f.g)(x) = 8x, domain: all real numbers.
b. (g.f)(x) = x^8, domain: all positive x values.
c. (f.h)(x) = 7ln(x), domain: all real numbers.
I hope this explanation helps you understand how to approach composition of functions! Let me know if you have any further questions.