Is Functions part of the EAPCET Mathematics syllabus?

Yes. Functions is taught in Intermediate first year · Paper IA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in Functions?

Forgetting that a logarithm in a denominator must also be non-zero; exclude the x where its argument equals 1. Reading g ∘ f as “apply g first”; g ∘ f(x) = g(f(x)), so f acts first. Calling a function onto by looking only at the formula; onto depends on the codomain, so compare the range with the stated codomain. Writing mⁿ instead of nᵐ for the number of functions from an m-element set to an n-element set; each of the m inputs has n choices, giving nᵐ.

EAPCET Mathematics · Intermediate first year · Paper IA

Functions

Q: How should I prepare Functions for EAPCET?

Domain questions are lost on carelessness: check every denominator, square root and logarithm separately, then intersect the conditions. Sketch modulus and greatest-integer functions instead of reasoning in your head — the graph usually shows the answer.

Functions sets the language for the rest of the course: what a function is, which inputs it accepts, what it can output, and when it can be reversed or combined with another function.

Subject: Mathematics
Taught in: Intermediate first year · Paper IA
Mathematics in the paper: 80 of 160 questions

Key formulas and results

A function has an inverse exactly when it is a bijection (one-one and onto).
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ for bijections $f$ and $g$ .
Number of functions from a set of $m$ elements to a set of $n$ elements $= n^{m}$ .
Number of one-one functions from $m$ elements to $n$ elements $(m \le n)$ $= {}^{n}P_{m}$ .
For $\sqrt{f(x)}$ the domain needs $f(x) \ge 0$ ; for $\tfrac{1}{f(x)}$ it needs $f(x) \ne 0$ .
$f$ is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$ .
Number of onto functions from a set of $m$ elements to a set of $n$ elements $(m \ge n)$ : $n^{m} - {}^{n}C_{1}(n - 1)^{m} + {}^{n}C_{2}(n - 2)^{m} - {}^{n}C_{3}(n - 3)^{m} + \dots$
Number of bijections between two sets of $n$ elements $= n!$ (and 0 if the sets have different sizes)
Number of many-one functions from $m$ elements to $n$ elements $= n^{m} - {}^{n}P_{m}$ when $m \le n$ , and all $n^{m}$ functions when $m > n$
Domain of $\log_{b} f(x)$ : $f(x) > 0$ , $b > 0$ and $b \ne 1$ ; domain of $\sin^{-1}(h(x))$ or $\cos^{-1}(h(x))$ : $-1 \le h(x) \le 1$
Period of $\sin px$ and $\cos px$ $= \dfrac{2\pi}{|p|}$ ; period of $|\sin px|$ and $|\cos px|$ $= \dfrac{\pi}{|p|}$
Range of a strictly increasing $f$ on $[a, b]$ is $[f(a), f(b)]$ ; of a strictly decreasing $f$ it is $[f(b), f(a)]$
Graph shifts: $y = f(x + a)$ moves $y = f(x)$ left by $a$ units $(a > 0)$ and $y = f(x) + a$ moves it up by $a$ units

What this chapter covers

Types of functions: one-one, onto and bijective
Domain and range of real-valued functions
Composite functions
Inverse of a function
Even and odd functions
Modulus and greatest integer functions

How it is usually asked

Find the domain or range of a given real function
Decide whether a function is one-one, onto or both
Compute a composite function or an inverse
Count the functions between two finite sets

Understanding Functions

A function f: A → B assigns to every element of A exactly one element of B. A is the domain, B is the codomain, and the set of values f actually produces is the range. f is one-one if different inputs always give different outputs, and onto if every element of B is produced by some input. A function that is both is a bijection, and only bijections have an inverse, because reversing the rule must again give exactly one output.

For a real function given by a formula, the domain is every x for which the formula makes sense: denominators must not be zero, expressions under a square root must be non-negative and the argument of a logarithm must be positive. The range is usually found by solving y = f(x) for x and asking which values of y give a valid x. The composite g ∘ f means apply f first and then g, so g(f(x)) needs f(x) to lie in the domain of g.

EAPCET questions on this chapter are short and direct: find a domain or range, decide whether a function is one-one or onto, compute f(f(x)) or an inverse, or count functions between finite sets. Counting results such as nᵐ functions and ⁿPₘ one-one functions turn the last type into a one-line answer, while domain questions reward listing every condition and intersecting them.

Worked example

EAPCET-style question

The domain of $f(x) = \sqrt{x - 1} + \dfrac{1}{\log(4 - x)}$ is

(1) $[1, 4)$
(2) $[1, 3) \cup (3, 4)$
(3) $(1, 4)$
(4) $[1, 3)$

Solution

The square root needs $x - 1 \ge 0$ , so $x \ge 1$ .
The logarithm needs $4 - x > 0$ , so $x < 4$ .
The denominator needs $\log(4 - x) \ne 0$ , so $4 - x \ne 1$ , that is $x \ne 3$ .
Intersecting the three conditions gives $[1, 3) \cup (3, 4)$ .

Answer: $[1, 3) \cup (3, 4)$

Shortcuts and results to remember

Horizontal line test: if some horizontal line cuts the graph more than once the function is many-one; if $f'(x) > 0$ throughout (or $f'(x) < 0$ throughout) it is one-one.
$y = |f(x)|$ is drawn by reflecting the part of $y = f(x)$ below the $x$ -axis upward, while $y = f(|x|)$ keeps the part for $x \ge 0$ and mirrors it in the $y$ -axis.
Standard functional equations: $f(xy) = f(x) + f(y)$ gives $f(x) = k\log x$ , $f(xy) = f(x)f(y)$ gives $f(x) = x^{n}$ , $f(x + y) = f(x) + f(y)$ gives $f(x) = x f(1)$ , and $f(x + y) = f(x)f(y)$ gives $f(x) = [f(1)]^{x}$ .
If a polynomial satisfies $f(x) + f\left(\tfrac{1}{x}\right) = f(x)\,f\left(\tfrac{1}{x}\right)$ , then $f(x) = 1 \pm x^{n}$ .
For domains involving $[x]$ , first solve the inequality for $t = [x]$ , then turn the allowed integer values of $t$ back into intervals of $x$ .

Common mistakes to avoid

Forgetting that a logarithm in a denominator must also be non-zero; exclude the x where its argument equals 1.
Reading g ∘ f as “apply g first”; g ∘ f(x) = g(f(x)), so f acts first.
Calling a function onto by looking only at the formula; onto depends on the codomain, so compare the range with the stated codomain.
Writing mⁿ instead of nᵐ for the number of functions from an m-element set to an n-element set; each of the m inputs has n choices, giving nᵐ.

How to prepare it

Domain questions are lost on carelessness: check every denominator, square root and logarithm separately, then intersect the conditions.

Sketch modulus and greatest-integer functions instead of reasoning in your head — the graph usually shows the answer.

Once the formulas feel familiar, test them the way the exam does: practise Functions questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Functions chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Functions part of the EAPCET Mathematics syllabus?: Yes. Functions is taught in Intermediate first year · Paper IA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Functions for EAPCET?: Domain questions are lost on carelessness: check every denominator, square root and logarithm separately, then intersect the conditions. Sketch modulus and greatest-integer functions instead of reasoning in your head — the graph usually shows the answer.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Functions?: Forgetting that a logarithm in a denominator must also be non-zero; exclude the x where its argument equals 1. Reading g ∘ f as “apply g first”; g ∘ f(x) = g(f(x)), so f acts first. Calling a function onto by looking only at the formula; onto depends on the codomain, so compare the range with the stated codomain. Writing mⁿ instead of nᵐ for the number of functions from an m-element set to an n-element set; each of the m inputs has n choices, giving nᵐ.