MathematicsChapter 6: Functions and Graphs
EVALUATING f(x)
Replace every x by the given number or expression.
Learning objectives
- Explain evaluating f(x) using the correct definitions, conditions, and notation.
- Apply replace every x, f(t+1) != f(t)+1 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Replace every x by the given number or expression. Brackets help: f(t+1) means substitute t+1 into the whole rule, not f(t)+1. Arithmetic with fractions stays fraction-accurate.
Formula
t = 3
Where / symbols
- EVALUATING f(x)
- Replace every x by the given number or expression. Brackets help: f(t+1) means substitute t+1 into the whole rule, not f(t)+1. Arithmetic with fractions stays fraction-accurate.
- NUMBERS
- If f(x)=x^2 - (1/2)x + 3, then f(2)=4-1+3=6. f(-1)=1+1/2+3=4.5. Keep the order: square first if the formula says x^2.
- EXPRESSIONS AND MODELS
- f(t+1)=(t+1)^2 - (1/2)(t+1)+3. A value model V(t)=9500-860t: V(5)=5200. If V(t)=6920 then 860t=2580, t=3.
Relationship
- f(t+1) != f(t)+1
Overview
Plug in carefully:
Numbers
If f(x)=x2 - (12)x + 3, then f(2)=4-1+3=6. f(-1)=1+12+3=4.5. Keep the order: square first if the formula says x^2.
Examples f(2)=6 · f(-1)=9/2 · f(2/3)=28/9
Expressions and Models
f(t+1)=(t+1)2 - (12)(t+1)+3. A value model V(t)=9500-860t: V(5)=5200. If V(t)=6920 then 860t=2580, t=3.
Examples V(5)=5200 · t=3 years · Not V(t)+1
WHAT — EVALUATING f(x)
Replace every x by the given number or expression. Brackets help: f(t+1) means substitute t+1 into the whole rule, not f(t)+1. Arithmetic with fractions stays fraction-accurate. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Evaluating F(X) matters because functions describe how one quantity depends on another. Domain, range, composition, and inverse ideas are the language used for graphs and mathematical models.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through NUMBERS, then EXPRESSIONS AND MODELS. Worked: f(x) = x^2 - (12)x + 3. Find f(2). f(2) = (2)2 - (12)(2) + 3 f(2) = 4 - 1 + 3 f(2) = 6 Worked: V(t) = 9500 - 860t. If V(t) = 6920, find t. 9500 - 860t = 6920 9500 - 6920 = 860t 2580 = 860t t = 3 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- f(x) = x2 - (12)x + 3. Find f(2).
- f(2) = (2)2 - (12)(2) + 3
- f(2) = 4 - 1 + 3
- f(2) = 6
- V(t) = 9500 - 860t. If V(t) = 6920, find t.
- 9500 - 860t = 6920
- 9500 - 6920 = 860t
- 2580 = 860t
- t = 3
In simple words
f(2) is a number. f is the rule. Do not write f=6.
Key points
- replace every x
- f(t+1) != f(t)+1
- BODMAS/PEMDAS
- whole substitute
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.EVALUATING f(x)
Sticky Notes · Quick Revision
Core Idea
- Replace every x by the given number or expression. Brackets help: f(t+1) means substitute t+1 into the whole rule, not f(t)+1. Arithmetic with fractions stays fraction-accurate.
NUMBERS
- If f(x)=x2 - (12)x + 3, then f(2)=4-1+3=6. f(-1)=1+12+3=4.5. Keep the order: square first if the formula says x^2.
EXPRESSIONS AND MODELS
- f(t+1)=(t+1)2 - (12)(t+1)+3. A value model V(t)=9500-860t: V(5)=5200. If V(t)=6920 then 860t=2580, t=3.
Worked Method
- Worked: f(x) = x2 - (12)x + 3. Find f(2).
- f(2) = (2)2 - (12)(2) + 3
- f(2) = 4 - 1 + 3
- f(2) = 6
- Worked: V(t) = 9500 - 860t. If V(t) = 6920, find t.
- 9500 - 860t = 6920
- 9500 - 6920 = 860t
- 2580 = 860t
- t = 3
Common Misconception
- f(2) is a number. f is the rule. Do not write f=6.
FBISE Exam Check
- f(2) is a number. f is the rule. Do not write f=6.
60-second revision
- Replace every x by the given number or expression.
- NUMBERS: If f(x)=x2 - (12)x + 3, then f(2)=4-1+3=6.
- EXPRESSIONS AND MODELS: f(t+1)=(t+1)2 - (12)(t+1)+3.
- Worked-method reminder: t = 3
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