Evaluating functions worksheet with answers and solutions

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

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

vs

g(w) + g(1) = w2 + 12 = w2 + 1

Different Result!

Questions 1-4 : Write the given equation as a function of x.

Question 1 :

2x + 3y - 5  =  0

Question 2 :

x2y + 3xy - 3 = 0

Question 3 :

log10y = x

Question 4 :

log(x) + log(y) = log (x + y)

Question 5 :

Evaluate f(4) where f(x) = 3(2x + 1).

Question 6 :

Evaluate f(w + 2) where f(x) = x2 + 3x + 5.

Question 7 :

Evaluate f(3) where f(m) = (2m2 + 5m - 7)/2.

Question 8 :

Given f = x2 - x - 4, if f(k) = 8, what is the value of k? 

1. Answer :

2x + 3y - 5 = 0

To write the given equation as a function of x, define the given equation by y in terms of x.

2x + 3y - 5 = 0

Subtract 2x and 5 from both sides.

3y = 5 - 2x

Divide both sides by 3.

y = (5 - 2x)/3

y = (5 - 2x)/3

Let y = f(x).

f(x) = (5 - 3x)/3

2. Answer :

x2y + 3xy - 3 = 0

Add 3 to both sides.

x2y + 3xy = 3

y(x2 + 3x) = 3

Divide both sides by (x2 + 3x).

y = 3/(x2 + 3x)

3. Answer :

log10y = x

Convert the equation to exponential.

y = 10x

Let y = f(x).

f(x) = 10x

4. Answer :

log(x) + log(y) = log(x + y)

log(xy) = log(x + y)

xy = x + y

Subtract y from both sides.

xy - y = x

y(x - 1) = x

Divide both sides by (x - 1).

y = x/(x - 1)

Let y = f(x).

f(x) = x/(x - 1)

5. Answer :

f(x) = 3(2x + 1)

Substitute x = 4. 

f(4) = 3[2(4) + 1]

= 3[8 + 1]

= 3(9)

= 27

6. Answer :

f(x) = x2 + 3x + 5

Substitute x = w + 2.

f(w + 2) = (w + 2)2 + 3(w + 2) + 5

= (w + 2)(w + 2) + 3w + 6 + 5

= w2 + 2w + 2w + 4 + 3w + 6 + 5

= w2 + 7w + 15

7. Answer :

f(m) = (2m2 + 5m - 7)/2

Substitute m = 3.

f(3) = [2(3)2 + 5(3) - 7]/2

= [2(9) + 15 - 7 ]/2

= (18 + 15 - 7)/2

= 26/2

= 13

8. Answer :

f(k) = 8

k2 - k - 4 = 8

Subtract 8 from both sides.

k2 - k - 12 = 0

factor and solve.

(k + 3)(x - 4) = 0

k + 3 = 0   or   k - 4 = 0

k = -3   or   k = 4

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