Non-Calc - Algebra

If a quadratic equation doesn’t factorise in the normal way, you should use the quadratic formula:

x = -b ± √b^2 - 4ac / 2a

With the equation being ax^2 + bx + c

This method is used for solving quadratics:

1 - Rearrange the quadratic equation into its standard format
2 - Write out the initial bracket of (x + b/2)^2
3 - Multiply out the brackets and compare to the original equation
4 - Subtract adjusting number
5 - To then solve the equation, put the the extra number in the other side
6 - The square root both sides
7 - Put b on the other side and then you have x = b plus or minus the square root of the extra number.

Inverse Functions:
The inverse of a function f(x) is another function, f^-1(x), which reverses f(x).
First you should write out the equation x=f(y) (which is the same as f(x), just with y’s instead of x’s.
Then you should rearrange the equation to make y the subject.
Finally, replace y with f^-1(x).

Example 1:
If f(x) = 12 + x / 3, find f^-1(x)
x = 12 + y / 3
3x = 12 + y
y = 3x - 12
f^-1(x) = 3x - 12

Example 2:
If f(x) = 3 / 2x+5, show that ff^-1(x) = x
Inverse function - f^-1 = 3/2x - 5/2
Substitute f^-1(x) into f(x)
ff^-1(x) = 3/2(3/2x - 5/2)+5
= 3/(3/x - 5 +5)
= 3/(3/x)
= 3 x x/3
= x

Combining Functions:
Two functions (e.g. f(x) and g(x)) can be combined into a single function called a composite function.
Composite functions are written like this, fg(x), for example which means to do g first, then f. The one closest to the function should be done first.
To find a composite function, rewrite fg(x) as f(g(x)) and then replace g(x) with the expression it represents and then put this into f.
Note - fg(x) does not = gf(x).

Example 2:
If f(x) = 2x - 10 and g (x) = -x/2, find fg(x)
fg(x) = f(g(x)) = f(-x/2) = 2(-x/2) - 10 = -x - 10

Example 3:
If f(x) = 3x^2 and g(x) = x -5, find gf(2)
gf(2) = 3(2)^2 - 5
We have substituted the x in g(x) for f(x) and them substituted the x in f(x) for 2.