Functions
A function is a rule that assigns exactly one output to each input. Functions are fundamental to all of mathematics and appear in aptitude exams in forms like f(x) notation, composite functions, and inverse functions.
Fundamental Principles
Function
A relation f from set A to set B where every element in A has exactly one image in B. Notation: f: A → B.
Domain and Range
Domain is the set of all valid inputs. Range is the set of all resulting outputs.
Composite Function
(f ∘ g)(x) = f(g(x)). Apply g first, then f to the result.
Inverse Function
f⁻¹ is the function that reverses f. If f(a) = b, then f⁻¹(b) = a. f⁻¹ exists only if f is one-one and onto.
Essential Formulation Tips
- Check domain restrictions: denominators ≠ 0, square roots ≥ 0.
- For composition (f ∘ g)(x), evaluate inner function first.
- To find f⁻¹(x), swap x and y in y = f(x) and solve for y.
- If f(f(x)) = x, then f is its own inverse.
Shortcut Execution Techniques
- Even function: f(-x) = f(x). Graph is symmetric about y-axis.
- Odd function: f(-x) = -f(x). Graph is symmetric about origin.
- f(x+a) shifts graph left by a; f(x-a) shifts right by a.
- For linear f: f(ax+b) replaces every x with (ax+b).
Contextual Inquiries (FAQs)
Q: What is the difference between a relation and a function?
A: Every function is a relation, but not every relation is a function. A function has exactly one output per input.
Q: How do I find the domain of f(x) = 1/(x-2)?
A: The denominator cannot be zero: x ≠ 2. Domain = all real numbers except 2.
Q: What does (f ∘ g)(x) mean?
A: It means f(g(x)) — first apply g to x, then apply f to the result.
Example Breakdown: Evaluating a Function
Most basic function problem type.f(2) = 2(4) - 3(2) + 1
= 8 - 6 + 1
= 3
Example Breakdown: Composite Function
Order matters: f(g(x)) ≠ g(f(x)) generally.g(4) = 8
f(g(4)) = f(8) = 8 + 3 = 11
Example Breakdown: Finding the Inverse
Standard inverse function technique.Let y = 3x - 5
Swap x and y: x = 3y - 5
Solve for y: y = (x + 5)/3
f⁻¹(x) = (x + 5)/3
Functions Practice Set 1
Basic function evaluation and domain-range identification.
Q1. If f(x) = 3x + 2, find f(5).
Q2. What is the domain of f(x) = 1/(x - 3)?
Q3. If f(x) = x² + 1, find f(-3).
Q4. What is the range of f(x) = x² for all real x?
Q5. If f(x) = |x|, what is f(-7)?
Q6. Which of the following is NOT a function?
Q7. If f(x) = 2x - 1 and f(a) = 7, find a.
Q8. What is the domain of f(x) = √(x - 4)?
Q9. If f(x) = x³, is f an odd or even function?
Q10. If g(t) = t² - 4t + 3, find g(3).