What to recall from Functions?

This note will focus on a quick summary of the Grade 11 Functions course, or MCR3UO.




First, a function is simply a relationship between an input and an output. This is usually expressed in the form of two variables, most commonly X and Y.


Example 1: y = x + 5


This is an example of a linear relationship where if y = x + 5 was plotted on a graph, it would form a line. If you need help with linear equations, feel free to check out the grade 9 math notes.


Example 2: y = x^2 -6x + 5


This is an example of a parabolic relationship. Unlike the first example, x is to the power of 2, which means that we are unable to figure out the roots of the equation simply by rearranging the equation. However, we can always use the quadratic formula to figure it out.


The roots are 1 and 5. If you are unable to get this, please review the grade 10/11 note on factoring.


To quickly review exponent laws,


Example 1: a x a x a = a ^ 3


This is because we simply ADD the exponents when we multiply them together, so, 2a^2 x a^3 = 2a^(2+3) or a^5) Remember constant multiples, which are the numbers that come before an exponent (if there is no number than it is a 1, i.e. a = 1a, and it can't equal 0a because 0 times anything equals 0), are just multipled with the other constant multiple normally.


Example 2: (999x) ^ 0 = 1


Don't make me explain this, just understand that anything to the power of 0 is one, including 0.0001 to the power of 0. EXCEPT 0 to the power 0, which does not exist



Example 2: x^-1 = (1/x^1).


This means that x to the power of negative one is the same as one over x to the power of one. To keep it simple, if you see a negative, bring it to the other side (i.e. Numerator to Denominator or Denominator to Numerator). So for a more complicated example, 5x^-3 / 2 would equal 5 / ((x^3) x 2), this is because the CONSTANT multiple is not affected by the negative, just the variable. So as well, (x^-1) / (x^-2) would be x^2 / x^1 which can simplify to just x.


Example 3: 2x^m x 3x^n = 6x^(m+n)


Simply add the exponents when multiplying two terms with the same base (i.e. in this case the base was x) REMEMBER that if the base was 3, i.e. 3^5 x 3^10, you still just add the exponents so the answer in this case would be 3^15. If you expand the numbers and multiply normaly, you should get the same answer.


Example 4: (2x^m) / (3x^n) = (2/3)(x)^(m/n)


Simply subtract the exponents when dividng two terms with th same base. Remember to watch for double negatives (i.e. (x^m)/(x^(-n)) = x^(m+n)


Example 5: (x^m)^n = x ^(m x n)

When a term has an exponent and than is put (or raised) to the exponent of something else, you just multiply the two exponents together.


Example 6: (x/y)^n = (x^n)/ (y^n)

The exponent affects both terms. In this case, x to the power of 1 was put (or raised)  to the exponent of n so it became x to the power of 1 times n.


Fractional Exponents : x ^ (m/n ) = (x ^ m) to the root of n.

i.e. x^(3/2) = (x^3) to the root of 2 (or square root)


Remember that if two terms have different exponents, they cannot be added or subtracted!

i.e. x^5 - x ^3 cannot be added! But what you can do in this case is factor out x^3 and than factor the remaining term, x^3 ( x^2 - 1) = (x^3)(x+1)(x-1)