f(x)=4/(x-4) and g(x)=1/x
The composition of f and g is written "fog".
(fog)(x)=f[g(x)]
This means use the function g(x) as the "input" on the function f(x).
(fog)(x)=f[1/x]
Sometimes it helps to use words when you are trying to input one function into another.
Think of "input" for "x" in f(x). Then f(x) would be "four divided by (input-4)".
Now for f[1/x], the "1/x" is the input, so put 1/x wherever you said "input" for f(x)
(fog)(x)=4/(1/x - 4)
Next step is to simplifying this. It helps to rewrite the complex fraction (a fraction that has a fraction in the numerator, denominator, or both) by taking out the bar (fraction line) and replacing it with a division symbol.
(fog)(x)=4 (division symbol) (1/x - 4)
Get common denominators for the 1/x and 4:
(fog)(x)=4 (division symbol) (1/x - 4x/x)
Denominators are alike, you can now subtract:
(fog)(x)=4 (division symbol) (1-4x)/x
Invert and multiply:
(fog)(x)=4/1 times x/(1/4x)
Multiply numerators and denominators:
(fog)(x)=4x/(1-4x)
Domains are always "All real numbers except ..."
The two exceptions are:
1. Whatever causes a zero in a denominator
2. Whatever causes a negative under an even indexed radical, (ex. square root)
So, you start domains with "all real numbers" and then throw out any numbers that cause "1" and "2" above.
For 4x/(1-4x) there is an unknown in the denominator, so something will have to be thrown out. If x were 1/4, then the denominator would be zero, so it is x=1/4 that has to be thrown out.
Domain, all real numbers except x=1/4.
In interval notation, this would be (-infinity, 1/4) U (1/4, + infinity)
To write interval notation, think of moving along the number line left to right.
Now, I didn't say anything different than Mr. Chu .. just expansions and explanations based on where students typically have difficulties when trying to do this.
