Maths1_Week5_Functions

Maths 1 Week 5: Functions (Inverse & Composite)

0. Prerequisites

NOTE

What you need to know:

• Function Notation: $f(x)$ means “plug $x$ into the machine $f$“.
• Solving for x: Rearranging equations like $y=2x+1\to x=(y-1)/2$.
• Domain: The set of allowed inputs (no division by zero, no negative roots).

Quick Refresher

• Even Function: Symmetric about Y-axis. $f(-x)=f(x)$ (e.g., $x^2$).
• Odd Function: Symmetric about Origin. $f(-x)=-f(x)$ (e.g., $x^3$).
• One-to-One: Each $y$ comes from only one $x$. (Passes Horizontal Line Test).

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1. Core Concepts

1.1 Composite Functions

• Definition: $(f\circ g)(x)=f(g(x))$.
• Meaning: Run input through $g$, then take the result and run it through $f$.
• Domain: $x$ must be in Domain of $g$, AND $g(x)$ must be in Domain of $f$.

1.2 Inverse Functions

• Definition ($f^{-1}$): The function that “undoes” $f$.
  • If $f(a)=b$, then $f^{-1}(b)=a$.
• Condition: Function must be Bijective (One-to-One and Onto).
• Graph: Reflection about the line $y=x$.

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2. Pattern Analysis & Goated Solutions

Pattern 1: Finding Domain of Composite Functions

Context: “Find the domain of $f(g(x))$.”

TIP

Mental Algorithm:

1. Inner Domain: Find domain of inner function $g(x)$. Call this $D_1$.
2. Outer Constraint: Find what inputs the outer function $f(u)$ accepts.
3. Solve Inequality: Set $g(x)$ inside the valid range of $f$. Solve for $x$. Call this $D_2$.
4. Intersection: Final Domain = $D_1\cap D_2$.

Example (Detailed Solution)

Problem: Let $f(x)=\sqrt{x}$ and $g(x)=x^2-4$. Find domain of $f(g(x))$. Solution:

1. Inner Domain ($g$):
   • $g(x)=x^2-4$. Polynomial.
   • $D_1=\mathbb{R}$ (All real numbers).
2. Outer Constraint ($f$):
   • $f(u)=\sqrt{u}$.
   • Requires $u\ge 0$.
3. Solve Inequality:
   • We need $g(x)\ge 0$.
   • $x^2-4\ge 0$.
   • $x^2\ge 4$.
   • $x\ge 2$ OR $x\le -2$.
   • $D_2=(-\infty ,-2]\cup [2,\infty )$.
4. Intersection:
   • $\mathbb{R}\cap D_2=D_2$. Answer: $(-\infty ,-2]\cup [2,\infty )$.

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Pattern 2: Finding the Inverse Function

Context: “Find $f^{-1}(x)$ for $f(x)=\frac{2x+1}{x-3}$.”

TIP

Mental Algorithm:

1. Swap: Replace $f(x)$ with $y$, then swap $x$ and $y$. ($x$ becomes $y$, $y$ becomes $x$).
2. Solve: Rearrange the equation to isolate the new $y$.
3. Replace: Call the new $y$ as $f^{-1}(x)$.

Example (Detailed Solution)

Problem: Find inverse of $f(x)=\frac{2x+1}{x-3}$. Solution:

1. Write as y: $y=\frac{2x+1}{x-3}$.
2. Swap: $x=\frac{2y+1}{y-3}$.
3. Solve for y:
   • Multiply by denominator: $x(y-3)=2y+1$.
   • Expand: $xy-3x=2y+1$.
   • Group $y$ terms: $xy-2y=3x+1$.
   • Factor $y$: $y(x-2)=3x+1$.
   • Divide: $y=\frac{3x+1}{x-2}$. Answer: $f^{-1}(x)=\frac{3x+1}{x-2}$.

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Pattern 3: Exponential Growth (Half-Life/Doubling)

Context: “Bacteria doubles every 3 hours”, “Element has half-life of 5 years”.

TIP

Mental Algorithm: Use the formula: $A(t)=A_0\cdot (Factor{)}^{t/Period}$.

• Doubling: Factor = 2. Period = Doubling time.
• Half-Life: Factor = 1/2. Period = Half-life time.

Example (Detailed Solution)

Problem: A population of 100 bacteria doubles every 4 hours. How many after 12 hours? Solution:

1. Identify:
   • $A_0=100$.
   • Factor = 2 (Doubles).
   • Period = 4 hours.
   • Time $t=12$.
2. Formula: $A(12)=100\cdot 2^{12/4}$.
3. Calculate:
   • $12/4=3$.
   • $A(12)=100\cdot 2^3$.
   • $A(12)=100\cdot 8=800$. Answer: 800 bacteria.

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3. Practice Exercises

1. Composite: $f(x)=x^2,g(x)=x+1$. Find $f(g(x))$.
   • Hint: $(x+1{)}^2=x^2+2x+1$.
2. Inverse: Find inverse of $y=3x-5$.
   • Hint: $x=3y-5⟹x+5=3y⟹y=(x+5)/3$.
3. Domain: Domain of $\sqrt{x^2}$.
   • Hint: $x^2\ge 0$ is always true. Domain is $\mathbb{R}$.

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🧠 Level Up: Advanced Practice

Question 1: Injectivity & Surjectivity

Problem: Check if $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^3+x$ is Bijective. Logic:

1. One-to-One (Injective):
   • $f^{\prime }(x)=3x^2+1$. Since $x^2\ge 0$, $f^{\prime }(x)\ge 1>0$.
   • Strictly increasing functions are always 1-1. Yes.
2. Onto (Surjective):
   • As $x\to \infty ,f(x)\to \infty$.
   • As $x\to -\infty ,f(x)\to -\infty$.
   • Polynomials of odd degree cover all $\mathbb{R}$. Yes. Answer: Bijective.

Question 2: Inverse of Restricted Function

Problem: Find inverse of $f(x)=x^2+2x$ for $x\ge -1$. Logic:

1. Set y: $y=x^2+2x$.
2. Complete Square: $y=(x+1{)}^2-1$.
3. Solve for x:
   • $y+1=(x+1{)}^2$.
   • $x+1=\pm \sqrt{y+1}$.
4. Use Constraint: Since $x\ge -1$, $x+1\ge 0$. Take positive root.
   • $x+1=\sqrt{y+1}⟹x=\sqrt{y+1}-1$. Answer: $f^{-1}(x)=\sqrt{x+1}-1$.

Question 3: Even/Odd Properties

Problem: Is $h(x)=\log (\frac{1-x}{1+x})$ Even, Odd, or Neither? Logic:

1. Test: Find $h(-x)$.
2. Calc: $h(-x)=\log (\frac{1-(-x)}{1+(-x)})=\log (\frac{1+x}{1-x})$.
3. Reciprocal Property: $\frac{1+x}{1-x}=(\frac{1-x}{1+x}{)}^{-1}$.
4. Log Property: $\log (A^{-1})=-\log (A)$.
5. Result: $h(-x)=-h(x)$. Answer: Odd Function.