Let ​f(x)=4−x and ​g(x)= 2 x. Perform the function operation and then find the domain of the result. ​(g−​f)(x)

Answer 1

Hey, I hope you’re doing well!

For this problem, we can start by finding (g-f)(x). Another way to write this is g(x)-f(x). So, we’ll subtract the two functions they gave us.

g(x)-f(x)

2x - (4-x) ← Parentheses, subtracting all of f(x)

2x - 4 + x ← After distributing the minus sign

3x - 4 ← After combining like terms, 2x + x

So, (g-f)(x) = 3x - 4.

Now to find the domain. The domain is what we can plug into the function (in this case, plug in for x). There are two rules of thumb for finding the domain.

Domain guidelines. Check for these first to see if there are domain restrictions.

1) Can’t have a negative number in a square root (which means we want zero or a positive number), and

2) Can’t divide by zero.

Since we don’t have a square root and also don’t have a fraction with the input variable (x, in this case) in the denominator, that means we can plug any value into the function (no restrictions on the domain). Which means the domain is all real numbers. Or, in interval notation: (-∞,∞).

I hope that helps! Take care!