Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska
📌 Based on National Curriculum 2023 / PECTAA 2026 Syllabus
📖 What's Inside: Complete review of Unit 4: MCQs on functions, composition, inverses; absolute value graphs; solving equations & inequalities; real-world applications (profit break-even, discount, GPS accuracy). Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 4: Functions and Graphs
Class 10 Math Unit 4 Review Exercise – Complete Revision
The Review Exercise of Unit 4 consolidates all concepts from Functions and Graphs – function operations, composition of functions (\( f \circ g \)), inverse functions (\( f^{-1} \)), absolute value functions, equations, and inequalities. This comprehensive review is designed to prepare students for Punjab Board exams with a mix of MCQs, short conceptual questions, and step-by-step solved problems covering every topic from Exercises 4.1 through 4.3.
What You Will Learn
This review exercise tests and reinforces all key concepts: function operations (addition, subtraction, multiplication, division), composite functions (\( f \circ g \)), inverse functions (\( f^{-1} \)), graphing absolute value functions with vertex and symmetry, solving absolute value equations and inequalities, and real-world applications.
Topics Covered in This Review
- MCQs – testing fundamental concepts of functions and composition
- Function Operations – addition, subtraction, multiplication, division
- Composite Functions – (\( f \circ g \)), (\( g \circ f \)), (\( f \circ f \)), (\( g \circ g \))
- Inverse Functions – finding \( f^{-1}(x) \) using the swap-and-solve method
- Absolute Value Graphs – graphing \( f(x) = |x| \) and transformations
- Absolute Value Equations – solving \( |ax+b| = c \)
- Absolute Value Inequalities – solving \( |ax+b| \le c \) and \( |ax+b| > c \)
- Real-World Applications – profit break-even, discounts, GPS accuracy
Why Review Exercise Is Important
The Review Exercise is the ultimate preparation tool for board exams. It combines all the skills learned in Exercises 4.1-4.3 and presents them in a format similar to what appears in board papers. Students who master this review exercise are well-prepared for any functions and graphs question in their exam.
Exam Tips for Functions and Graphs
- Remember: (\( f \circ g \))(x) = f(g(x)) – apply g first, then f
- For inverse functions, swap x and y, then solve for y
- Absolute value graphs are V-shaped with vertex at the point where the inside expression equals zero
- Always check domain restrictions for division and square root functions
📖 Multiple Choice Questions (Unit 4 Review)
1. If \( f(x) = \frac{5x-6}{3} \), then \( f(3) = \)
✅ Correct Answer: (B) 3
\( f(3) = \frac{5(3)-6}{3} = \frac{15-6}{3} = \frac{9}{3} = 3 \)
\( f(3) = \frac{5(3)-6}{3} = \frac{15-6}{3} = \frac{9}{3} = 3 \)
2. A function \( f \) from \( X \) to \( Y \) is represented by:
✅ Correct Answer: (C) \( f:X \to Y \)
Standard notation for a function from set X to set Y is \( f:X \to Y \).
Standard notation for a function from set X to set Y is \( f:X \to Y \).
3. \( (f \circ g)(x) = \)
✅ Correct Answer: (C) \( f(g(x)) \)
\( (f \circ g)(x) = f(g(x)) \) by definition of composite functions.
\( (f \circ g)(x) = f(g(x)) \) by definition of composite functions.
4. If \( f(x)=2x+3, g(x)=x+1 \), then \( f(x)+g(x)= \)
✅ Correct Answer: (B) \(3x+4\)
\( f(x)+g(x) = (2x+3)+(x+1) = 3x+4 \)
\( f(x)+g(x) = (2x+3)+(x+1) = 3x+4 \)
5. If \( f(x)=5x+2, h(x)=2x-2 \), then \( f(x)-h(x)= \)
✅ Correct Answer: (C) \(3x+4\)
\( f(x)-h(x) = (5x+2)-(2x-2) = 3x+4 \)
\( f(x)-h(x) = (5x+2)-(2x-2) = 3x+4 \)
6. If \( f(x)=3x+1, g(x)=2x \), then \( g(x) \times f(x)= \)
✅ Correct Answer: (D) \(6x^2+2x\)
\( g(x) \times f(x) = 2x(3x+1) = 6x^2+2x \)
\( g(x) \times f(x) = 2x(3x+1) = 6x^2+2x \)
7. If \( f(x)=x^2-4, g(x)=x+2, x\neq -2 \), then \( \frac{f(x)}{g(x)}= \)
✅ Correct Answer: (D) \(x-2\)
\( \frac{f(x)}{g(x)} = \frac{x^2-4}{x+2} = \frac{(x-2)(x+2)}{x+2} = x-2, x\neq -2 \)
\( \frac{f(x)}{g(x)} = \frac{x^2-4}{x+2} = \frac{(x-2)(x+2)}{x+2} = x-2, x\neq -2 \)
8. Shape of absolute value function graph?
✅ Correct Answer: (B) V-shaped
The absolute value function \( f(x) = |x| \) has a V-shaped graph.
The absolute value function \( f(x) = |x| \) has a V-shaped graph.
9. Vertical line test for a function: every vertical line intersects at:
✅ Correct Answer: (D) 1 point
The vertical line test states that a graph represents a function if every vertical line intersects the graph at most once.
The vertical line test states that a graph represents a function if every vertical line intersects the graph at most once.
10. If \( f(x)=x^3 \), then \( f(-2)= \)
✅ Correct Answer: (A) \(-8\)
\( f(-2) = (-2)^3 = -8 \)
\( f(-2) = (-2)^3 = -8 \)
2 If \( f(x)=25-x^2 \) and \( g(x)=5+x \), find:
(i) \( f(x)+g(x) \)
\[
\begin{aligned}
f(x)+g(x) &= (25-x^2)+(5+x) \\
&= -x^2+x+30
\end{aligned}
\]
(ii) \( f(x)-g(x) \)
\[
\begin{aligned}
f(x)-g(x) &= (25-x^2)-(5+x) \\
&= -x^2-x+20
\end{aligned}
\]
(iii) \( f(x)\cdot g(x) \)
\[
\begin{aligned}
f(x)\cdot g(x) &= (25-x^2)(5+x) \\
&= -x^3-5x^2+25x+125
\end{aligned}
\]
(iv) \( \frac{f(x)}{g(x)} \)
\[
\begin{aligned}
\frac{f(x)}{g(x)} &= \frac{25-x^2}{5+x} = \frac{(5-x)(5+x)}{5+x} \\
&= 5-x,\quad x \neq -5
\end{aligned}
\]
(v) \( f(7) \)
\[
f(7) = 25-49 = -24
\]
(vi) \( g(-8) \)
\[
g(-8) = 5-8 = -3
\]
3 If \( f(x)=x^3 \) and \( g(x)=14+2x \), find:
(i) \( (f \circ g)(x) \)
\[
(f \circ g)(x) = f(g(x)) = f(14+2x) = (14+2x)^3
\]
(ii) \( (g \circ f)(x) \)
\[
(g \circ f)(x) = g(f(x)) = g(x^3) = 14+2x^3
\]
(iii) \( (f \circ f)(x) \)
\[
(f \circ f)(x) = f(f(x)) = f(x^3) = (x^3)^3 = x^9
\]
(iv) \( (g \circ g)(x) \)
\[
(g \circ g)(x) = g(g(x)) = g(14+2x) = 14+2(14+2x) = 4x+42
\]
4 Find \( f^{-1}(x) \)
(i) \( f(x)=9x-1 \)
\[
y=9x-1 \implies x=\frac{y+1}{9} \implies f^{-1}(x)=\frac{x+1}{9}
\]
(ii) \( f(x)=\frac{5}{x-1},\; x\neq 1 \)
\[
y=\frac{5}{x-1} \implies x=\frac{5+y}{y} \implies f^{-1}(x)=\frac{5+x}{x}
\]
(iii) \( f(x)=\sqrt{x-5},\; x\ge 5 \)
\[
y=\sqrt{x-5} \implies x=y^2+5 \implies f^{-1}(x)=x^2+5,\; x\ge 0
\]
(iv) \( f(x)=\frac{3-x}{2} \)
\[
y=\frac{3-x}{2} \implies x=3-2y \implies f^{-1}(x)=3-2x
\]
5 Graph absolute value functions
(i) \( f(x)=7|x| \)
Vertex at \((0,0)\), V-shaped symmetric.
Vertex at \((0,0)\), V-shaped symmetric.
(ii) \( f(x)=|x+6|-2 \)
Vertex at \((-6,-2)\), opens upward.
Vertex at \((-6,-2)\), opens upward.
6 Solve absolute value equations
(i) \( |3x-2|=1 \)
\[
3x-2=1 \implies x=1 \quad \text{or} \quad 3x-2=-1 \implies x=\frac{1}{3}
\]
Solution: \( \left\{\frac{1}{3},\;1\right\} \)
(ii) \( |6x+1|=9 \)
\[
6x+1=9 \implies x=\frac{4}{3} \quad \text{or} \quad 6x+1=-9 \implies x=-\frac{5}{3}
\]
Solution: \( \left\{-\frac{5}{3},\;\frac{4}{3}\right\} \)
7 Solve absolute value inequalities
(i) \( |7-2x| \le 1 \)
\[
-1 \le 7-2x \le 1 \implies 3 \le x \le 4
\]
Solution: \([3,4]\)
(ii) \( |6x+18| \le 24 \)
\[
-24 \le 6x+18 \le 24 \implies -7 \le x \le 1
\]
Solution: \([-7,1]\)
8 \( f(x)=3x+7 \): inverse and solve
(i) \( f^{-1}(x) \)
\[
y=3x+7 \implies x=\frac{y-7}{3} \implies f^{-1}(x)=\frac{x-7}{3}
\]
(ii) Solve \( f(x)=f^{-1}(x) \)
\[
3x+7=\frac{x-7}{3} \implies 9x+21=x-7 \implies 8x=-28 \implies x=-\frac{7}{2}
\]
9 Break-even analysis
\( P(x)=100x-10000 \). How many items to break even?
\[
P(x)=0 \implies 100x-10000=0 \implies x=100
\]
✅ Must sell 100 items to break even.
10 Discount function
\( D(p)=0.85p \). Selling price for Rs. 2000?
\[
D(2000)=0.85(2000)=1700
\]
✅ Selling price: Rs. 1,700
11 GPS accuracy
\( |100-r| \le 6 \). Range of acceptable reported locations?
\[
-6 \le 100-r \le 6 \implies 94 \le r \le 106
\]
✅ Acceptable reported location \( r \) in \([94,\;106]\) metres.
📈 Key Formulas – Unit 4 Summary
- Composite Function: \( (f \circ g)(x) = f(g(x)) \)
- Inverse Function: \( f(f^{-1}(x)) = f^{-1}(f(x)) = x \)
- Absolute Value Equation: \( |X| = a \Rightarrow X = \pm a \) (if \( a\ge0 \))
- Absolute Value Inequality: \( |X| < a \Rightarrow -a < X < a \); \( |X| > a \Rightarrow X < -a \text{ or } X > a \)
- Absolute Value Graph: \( f(x) = |x-h| + k \) has vertex \((h,k)\)
❓ Frequently Asked Questions
What is covered in Unit 4 Review Exercise?
The Review Exercise covers all concepts from Unit 4 including MCQs on functions and composition, function operations, composite functions (\( f \circ g \)), inverse functions \( f^{-1}(x) \), graphing absolute value functions, solving absolute value equations and inequalities, and real-world applications like profit break-even, discounts, and GPS accuracy.
Is this review exercise important for board exams?
Yes, the review exercise consolidates all key concepts from Unit 4 and is an excellent preparation tool for Punjab Board exams. The MCQs and short questions are representative of what appears in board papers.
Who prepared these review solutions?
These solutions were prepared by Muhammad Tayyab, Subject Specialist Mathematics at Govt Christian High School Daska, for Hira Science Academy.