Evaluating Functions

f is just a name,
x is just a place-holder.

These are all the same function:

f(x) = 1 − x + x 2
f(q) = 1 − q + q 2
w(A) = 1 − A + A 2
pumpkin(θ) = 1 − θ + θ 2

Evaluate For a Given Value:

Let us evaluate that function for x=3 :

f( 3 ) = 1 − 3 + 3 2 = 1 − 3 + 9 = 7

Evaluate For a Given Expression:

Evaluating can also mean replacing with an expression (such as 3m+1 or v 2 ).

Let us evaluate our function for x=1/r :

f( 1/r ) = 1 − ( 1/r ) + ( 1/r ) 2

f( a−4 ) = 1 − ( a−4 ) + ( a−4 ) 2 = 1 − a + 4 + a 2 − 8a + 16 = 21 − 9a + a 2

Another Example

You can use your ability to evaluate functions in other way:

Example: h(x) = 3x 2 + ax − 1

You are told that h(3) = 8, can you work out what "a" is?

First, evaluate h(3): h( 3 ) = 3×( 3 ) 2 + a× 3 − 1 Simplify: h(3) = 27 + 3a − 1 h(3) = 26 + 3a Now . we are told that h(3) = 8: 8 = 26 + 3a Swap sides: 26 + 3a = 8 Subtract 26 from both sides: 3a = −18 Divide by 3: a = −6

Check: h(3) = 3(3) 2 − 6×3 − 1 = 27 − 18 − 1 = 8

Careful!

I recommend putting the substituted values inside parentheses () , so you don't make mistakes.

Example: evaluate the function h(x) = x 2 + 2 for x = −3

Replace the variable "x" with "−3":

h(−3) = (−3) 2 + 2 = 9 + 2 = 11

Without the () you could make a mistake:

h(−3) = −3 2 + 2 = −9 + 2 = −7 (WRONG!)

Also be careful of this:

f(x+a) is not the same as f(x) + f(a)