Evaluating Functions

To evaluate a function is to:

Replace (substitute) its variable with a given number or expression.

Like in this example:

Example: evaluate the function f(x) = 2x+4 for x=5

Just replace the variable "x" with "5":

f(5) = 2×5 + 4 = 14

Answer: f(5) = 14

More Examples

Here is a function:

f(x) = 1 − x + x²

Important! The "x" is just a place-holder! And "f" is just a name.

These are all the same function:

f(x) = 1 − x + x²
f(q) = 1 − q + q²
w(A) = 1 − A + A²
pumpkin(θ) = 1 − θ + θ²

Evaluate For a Given Value:

Let us evaluate that function for x=3:

f(3) = 1 − 3 + 3² = 1 − 3 + 9 = 7

Evaluate For a Given Expression:

Evaluating can also mean replacing with an expression (such as 3m+1 or v²).

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

f(1/r) = 1 − (1/r) + (1/r)²

Or evaluate the function for x = a−4:

f(a−4) = 1 − (a−4) + (a−4)²

= 1 − a + 4 + a² − 8a + 16

= 21 − 9a + a²

Another Example

You can use your ability to evaluate functions to find other answers:

Example: h(x) = 3x² + 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 know that h(3) = 8, so: 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 yes

Careful!

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

Example: evaluate the function h(x) = x² + 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) = −3²+ 2 = −9 + 2 = −7 (WRONG!)

Also be careful of this:

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

Example: g(x) = x²

g(w+1) = (w+1)² = w² + 2w + 1

g(w) + g(1) = w² + 1² = w² + 1

Different Result!

Evaluating Functions