Question

For f(x)=3x+1 and g(x)=x squared - 6 find (f- g)(x)

Answer 1

It helps to first clarify that the notation (f - g)(x) simply means f(x) - g(x). Given that, let's look at our f(x) and our g(x) here, and use their definitions to find their difference.

$f(x)=3x+1\\g(x)=x^2-6$

When we're taking (f - g)(x), we simply substitute the expression 3x + 1 for f(x) and the expression x² - 6 for g(x) to obtain:

$(3x+1)-(x^2-6)=3x+1-x^2+6$

Or, ordering the polynomial from highest power to lowest and combining the constants:

$-x^2+3x+7$

Edit: By request, here's what would happen if you had something instead like:

$(f\times g)(x)$

In this case, you'd have to *multiply* the two function expressions together. Here's what that would look like:

$(3x+1)(x^2-6)$

Using the distributive property, we can distribute the expression $3x+1$ to the terms $x^{2}$ and $-6$:

$(3x+1)x^2-(3x+1)6$

Distributing again, we get:

$3x(x^2)+x^2-3x(6)+6=3x^3+x^2-18x+6$

And we're done.