# Evaluating Functions

## 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 + x^{2}

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

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 the function for x=1/r:

f(1/r) = 1 − (1/r) + (1/r)^{2}

Or evaluate the function for x = a−4:

^{2}

^{2}− 8a + 16

^{2}

## Another Example

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

### Example: h(x) = 3x^{2} + ax − 1

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

^{2}+ a×3 − 1

**h(3) = 8**, so: 8 = 26 + 3a

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**

^{2}+ 2

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)**

### Example: g(x) = x^{2}

g(w+1) = (w+1)^{2} = **w ^{2} + 2w + 1**

*vs*

g(w) + g(1) = w^{2} + 1^{2} = **w ^{2} + 1**

Different Result!