For the given function f,g, and h, find f*g*h and state the exact domain of f*g*h
f(x)=e^x
g(x)=log_x
h(x)=x/x+8
I am not sure what you mean by
g(x) = log_x
If your g(x) is log(base e)(x) = ln x,
then
f*g(x) = e^(lnx) = x
and
f*g*h(x) = h(x) = x/(x+8)
However, x/(x+8) must be positive for g(x) to be defined.
Therefore x> 0 or x < -8 is the domain
i thinkthe only ans is x> 0,its impossible to log a negative no.u may try to use a calculator to prove it. then x/x should be done first.so x < -8 or x> -8 is also possible.
To find f*g*h, we need to multiply the functions f(x) = e^x, g(x) = log_x, and h(x) = x / (x + 8).
Step 1: Multiply f(x) and g(x).
(f*g)(x) = f(x) * g(x) = (e^x) * (log_x).
Step 2: Multiply the result of step 1 with h(x).
(f*g*h)(x) = (f*g)(x) * h(x) = [(e^x) * (log_x)] * (x / (x + 8)).
Now, let's simplify this expression further.
Step 3: Use the properties of logarithms to simplify (e^x) * (log_x).
We know that log_a(a^c) = c, so (log_x)(e^x) = x.
So, (f*g)(x) = x.
Step 4: Substitute the result of step 3 into (f*g*h)(x).
(f*g*h)(x) = [(x) * (x / (x + 8))] = x^2 / (x + 8).
Therefore, f*g*h(x) = x^2 / (x + 8).
Now, let's discuss the exact domain of f*g*h.
The domain depends on the original functions.
For f(x) = e^x, the domain is all real numbers since e^x is defined for any real number x.
For g(x) = log_x, the domain is positive real numbers since the logarithm function is defined for positive inputs.
For h(x) = x / (x + 8), the domain is all real numbers except x = -8, as the denominator cannot be zero.
Combining these domains, the exact domain of f*g*h is all real numbers except x = -8.