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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