MathematicsChapter 6: Functions and Graphs
Square Root, Reciprocal, Exponential (named)
sqrt(x): domain x>=0, range y>=0.
Learning objectives
- Explain square root, reciprocal, exponential (named) using the correct definitions, conditions, and notation.
- Apply sqrt domain x>=0, 1x undefined at 0 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
sqrt(x): domain x>=0, range y>=0. 1x: undefined at 0, axes are asymptotes, hyperbola. Exponential 2^x (and e^x, (13)^x): variable in the exponent. These are named families in Grade 10 graphs, not a full Class 11 exponential chapter.
Where / symbols
- SQUARE ROOT, RECIPROCAL, EXPONENTIAL (NAMED)
- sqrt(x): domain x>=0, range y>=0. 1/x: undefined at 0, axes are asymptotes, hyperbola. Exponential 2^x (and e^x, (1/3)^x): variable in the exponent. These are named families in Grade 10 graphs, not a full Class 11 exponential chapter.
- SQRT AND RECIPROCAL
- sqrt starts at the origin and rises slowly. 1/x has two branches. As x gets large positive, 1/x goes toward 0. Reciprocal of 0 is undefined, not a real number.
- EXPONENTIAL
- 2^x grows; (1/3)^x decays as x increases. They never output 0. Horizontal asymptote y=0 is the school picture for a^x with a>0, a!=1.
Relationship
- sqrt domain x>=0
Overview
Three more graphs:
Sqrt and Reciprocal
sqrt starts at the origin and rises slowly. 1x has two branches. As x gets large positive, 1x goes toward 0. Reciprocal of 0 is undefined, not a real number.
Examples sqrt(x) >= 0 · 1/x -> 0 for large x · No y for x=0 on 1/x
Exponential
2^x grows; (13)^x decays as x increases. They never output 0. Horizontal asymptote y=0 is the school picture for a^x with a>0, a!=1.
Examples 2^x · e^x named · (1/3)^x decays
WHAT — SQUARE ROOT, RECIPROCAL, EXPONENTIAL (NAMED)
sqrt(x): domain x>=0, range y>=0. 1x: undefined at 0, axes are asymptotes, hyperbola. Exponential 2^x (and e^x, (13)^x): variable in the exponent. These are named families in Grade 10 graphs, not a full Class 11 exponential chapter. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Square Root, Reciprocal, Exponential (Named) 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 SQRT AND RECIPROCAL, then EXPONENTIAL. Worked: domain and range of f(x) = sqrt(x). inside the square root must be >= 0 domain: x >= 0 principal square root is >= 0 range: y >= 0 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- domain and range of f(x) = sqrt(x).
- inside the square root must be >= 0
- domain: x >= 0
- principal square root is >= 0
- range: y >= 0
In simple words
x^2 is a polynomial (power on x). 2^x is exponential (power is x). Do not swap the names.
Key points
- sqrt domain x>=0
- 1x undefined at 0
- asymptotes x=0, y=0
- variable in the exponent
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Square Root, Reciprocal, Exponential (named)
Sticky Notes · Quick Revision
Core Idea
- sqrt(x): domain x>=0, range y>=0. 1x: undefined at 0, axes are asymptotes, hyperbola. Exponential 2^x (and e^x, (13)^x): variable in the exponent. These are named families in Grade 10 graphs, not a full Class 11 exponential chapter.
SQRT AND RECIPROCAL
- sqrt starts at the origin and rises slowly. 1x has two branches. As x gets large positive, 1x goes toward 0. Reciprocal of 0 is undefined, not a real number.
EXPONENTIAL
- 2^x grows; (13)^x decays as x increases. They never output 0. Horizontal asymptote y=0 is the school picture for a^x with a>0, a!=1.
Worked Method
- Worked: domain and range of f(x) = sqrt(x).
- inside the square root must be >= 0
- domain: x >= 0
- principal square root is >= 0
- range: y >= 0
Common Misconception
- x^2 is a polynomial (power on x). 2^x is exponential (power is x). Do not swap the names.
FBISE Exam Check
- x^2 is a polynomial (power on x). 2^x is exponential (power is x). Do not swap the names.
60-second revision
- sqrt(x): domain x>=0, range y>=0.
- SQRT AND RECIPROCAL: sqrt starts at the origin and rises slowly.
- EXPONENTIAL: 2x grows; (13)x decays as x increases.
- Worked-method reminder: range: y >= 0
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