given functions f,g, and h find f o g o h state exact domain. show work
f(x)=1/x g(x)=lnx h(x)=2x+15
h has domain all reals
ln(n) has domain n > 0
so, we need 2x+15 > 0 or x > -7.5
1/x has domain x not zero.
Since ln(u) = 0 when u = 1, we need g(h) not 1.
So, ln(2x+15) cannot be 0.
That is, 2x+15 cannot be 1.
so, x cannot be -7
The domain of fogoh is thus all reals greater than -7.5 except -7
thank you my precalculus is now done - chemistry is next- can't wait!!
To find the composition of functions f o g o h, we need to substitute the innermost function into the next inner function, and so on, until we reach the outermost function.
In this case, we start with h(x) = 2x + 15:
g o h = g(h(x)) = g(2x+15)
Next, we substitute g(x) = ln(x) into g(2x+15):
g(2x+15) = ln(2x+15)
Now, we can move to the next composition:
f o g o h = f(g o h) = f(ln(2x+15))
Finally, we substitute f(x) = 1/x into f(ln(2x+15)):
f(ln(2x+15)) = 1/(ln(2x+15))
The exact domain of the composition is the set of all x values that make the expression 2x+15 positive, since we cannot take the logarithm of a negative number or zero. Therefore, the domain is:
2x + 15 > 0
2x > -15
x > -15/2
So, the exact domain of f o g o h is x > -15/2.
To summarize:
f o g o h = 1/(ln(2x+15))
Domain: x > -15/2