Functions and Graphs: Types, Domains, Graphs, and Applications

1. Introduction to Functions

1.1 Definition of a Function

A function $f$ from a set $X$ to a set $Y$ is a rule that assigns to each element $x$ in $X$ exactly one element $y$ in $Y$. This is written as: $$y=f(x)$$ where:

• $x$ is the independent variable (input),
• $y$ is the dependent variable (output),
• $X$ is the domain (set of all possible inputs),
• $Y$ is the codomain (set of all possible outputs).

Key Points:

• A function must have exactly one output for each input (vertical line test).
• Notation: $f:X\to Y$ or $y=f(x)$.

1.2 Types of Functions

1.3 Domain and Range

• Domain: All possible input values ($x$) for which $f(x)$ is defined.
  • For $f(x)=\frac{1}{x}$, domain is $x\ne 0$.
  • For $f(x)=\sqrt{x}$, domain is $x\ge 0$.
• Range: All possible output values ($y$) that $f(x)$ can take.
  • For $f(x)=x^2$, range is $y\ge 0$.
  • For $f(x)=\sin (x)$, range is $[-1,1]$.

Worked Example 1: Find the Domain Find the domain of $f(x)=\frac{\sqrt{x-1}}{x-3}$. Solution:

1. The expression under the square root must be non-negative: $x-1\ge 0⟹x\ge 1$.
2. The denominator cannot be zero: $x-3\ne 0⟹x\ne 3$. Domain: $[1,3)\cup (3,\infty )$.

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2. Graphs of Functions

2.1 Plotting Basic Functions

Graphs visually represent the relationship between $x$ and $y=f(x)$. Key features to plot:

• Intercepts: $x$-intercepts ($f(x)=0$), $y$-intercepts ($x=0$).
• Symmetry: Even ($f(-x)=f(x)$) or odd ($f(-x)=-f(x)$).
• Asymptotes: Vertical ($x=a$), horizontal ($y=b$), or oblique.

Worked Example 2: Sketch the Graph Sketch the graph of $f(x)=|x|$. Solution:

• For $x\ge 0$, $f(x)=x$ (straight line with slope 1).
• For $x<0$, $f(x)=-x$ (straight line with slope -1).
• The graph forms a "V" with the vertex at $(0,0)$.

2.2 Transformations of Functions

Transformations shift, stretch, or reflect graphs of functions. Common transformations:

Worked Example 3: Apply Transformations Sketch the graph of $f(x)=(x-1{)}^2+2$. Solution:

1. Start with $f(x)=x^2$ (parabola opening upwards).
2. Shift right by 1: $(x-1{)}^2$.
3. Shift up by 2: $(x-1{)}^2+2$. Graph: Vertex at $(1,2)$, same shape as $x^2$.

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3. Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Example: $$f(x)=\{\begin{array}{ll}x+2& \text{if }x<0\\ 1-x& \text{if }x\ge 0\end{array}$$

Worked Example 4: Evaluate Piecewise Function Given $f(x)$ as above, find $f(-1)$ and $f(3)$. Solution:

• For $f(-1)$: Since $-1<0$, use $f(x)=x+2$. $f(-1)=-1+2=1$.
• For $f(3)$: Since $3\ge 0$, use $f(x)=1-x$. $f(3)=1-3=-2$.

Graphing Piecewise Functions:

1. Plot each piece separately.
2. Use open/closed circles to indicate exclusivity/inclusivity at boundaries.
3. Ensure continuity (if required) at transition points.

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4. Applications of Functions

4.1 Modeling Real-World Problems

Functions model relationships in physics, economics, and engineering. Examples:

1. Temperature vs. Height: Given ground temperature $20^{\circ }C$ and temperature at $1\text{ km}$ is $10^{\circ }C$, express $T(h)$ as a linear function. Solution: Slope $m=\frac{10-20}{1-0}=-10$. Equation: $T(h)=20+(-10)h=20-10h$.

2. Position Function from Acceleration: Given $a(t)=6t^2+1$, initial velocity $v(0)=4\text{ m/s}$, and initial displacement $s(0)=5\text{ cm}$, find $s(t)$. Solution:

   • Integrate acceleration to get velocity: $v(t)=\int a(t)dt=\int (6t^2+1)dt=2t^3+t+C$. Use $v(0)=4$: $4=0+0+C⟹C=4$. So, $v(t)=2t^3+t+4$.
   • Integrate velocity to get position: $s(t)=\int v(t)dt=\int (2t^3+t+4)dt=\frac{1}{2}{t}^4+\frac{1}{2}{t}^2+4t+D$. Use $s(0)=5\text{ cm}=0.05\text{ m}$: $0.05=0+0+0+D⟹D=0.05$. Final position function: $s(t)=\frac{1}{2}{t}^4+\frac{1}{2}{t}^2+4t+0.05$.

4.2 Optimization Problems

Functions are used to optimize quantities like cost, area, or volume. Example: Cost Minimization A rectangular storage container has volume $20{\text{ m}}^3$, length twice the width. Base costs Rs. 10/m², sides Rs. 4/m². Find dimensions to minimize cost. Solution:

1. Let width = $x$, length = $2x$, height = $h$.
2. Volume constraint: $x\cdot 2x\cdot h=20⟹h=\frac{20}{2x^2}=\frac{10}{x^2}$.
3. Cost function:
   • Base area: $2x^2$, cost: $10\cdot 2x^2=20x^2$.
   • Side areas: Two sides of $2x\cdot h$ (cost $4\cdot 2x\cdot h$) and two sides of $x\cdot h$ (cost $4\cdot x\cdot h$). Total side cost: $4(4xh+2xh)=4(6xh)=24xh$.
   • Total cost $C(x)=20x^2+24xh=20x^2+24x\cdot \frac{10}{x^2}=20x^2+\frac{240}{x}$.
4. To minimize $C(x)$, take derivative and set to zero: $C^{\prime }(x)=40x-\frac{240}{x^2}$. Set $C^{\prime }(x)=0$: $40x=\frac{240}{x^2}⟹40x^3=240⟹x^3=6⟹x=\sqrt[3]{6}$. Then $h=\frac{10}{(\sqrt[3]{6}{)}^2}$.

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5. Common Mistakes and Clarifications

5.1 Misconceptions

• Vertical Line Test: A graph represents a function only if no vertical line intersects it more than once.
• Domain vs. Range: Domain is about inputs ($x$), range is about outputs ($y$).
• Piecewise Functions: Forgetting to check the condition for each piece can lead to incorrect evaluations.

5.2 Comparing Functions

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Exam Tip

What to Expect in TU Exams

1. Definition-Based Questions:

   • Expect questions on defining functions, domains, ranges, and continuity.
   • Example: "Define continuity at a point $x=a$."

2. Graph Sketching:

   • Sketch graphs of piecewise, absolute value, and polynomial functions.
   • Label intercepts, asymptotes, and key points.

3. Function Evaluation:

   • Evaluate piecewise functions at specific points (e.g., $f(-1)$, $f(3)$).
   • Example: Given $f(x)=\{\begin{array}{ll}{x}^2& \text{if }x\le 0\\ 2x+1& \text{if }x>0\end{array}$, find $f(-2)$ and $f(1)$.

4. Applications:

   • Model real-world scenarios using linear or polynomial functions.
   • Example: Temperature variation with height, cost optimization.

5. Transformations:

   • Identify and apply transformations (shifts, stretches, reflections) to graphs.
   • Example: Given $f(x)=\sqrt{x}$, sketch $f(x-2)+3$.

6. Domain and Range:

   • Find domains of functions involving roots, denominators, and logarithms.
   • Example: Find the domain of $f(x)=\frac{\ln (x)}{x-1}$.

How to Score Full Marks

• Precision: Clearly state definitions and conditions (e.g., "domain excludes $x=0$").
• Graphs: Label axes, intercepts, and key features. Use dotted lines for asymptotes.
• Steps: Show all steps in calculations (e.g., integrating acceleration to velocity).
• Units: Include units in applied problems (e.g., meters, degrees Celsius).
• Verification: Check continuity at piecewise boundaries (e.g., ${\lim }_{x\to 0^{-}}f(x)={\lim }_{x\to 0^{+}}f(x)$).

Practice Questions for Revision

1. Find the domain and range of $f(x)=\frac{\sqrt{4-x^2}}{x-1}$.
2. Sketch the graph of $f(x)=\{\begin{array}{ll}{x}^2& \text{if }x\le 1\\ 2x-1& \text{if }x>1\end{array}$.
3. A particle's acceleration is $a(t)=3t+2$. If its initial velocity is $5\text{ m/s}$ and initial position is $0$, find its position function $s(t)$.
4. Express the temperature $T$ in ${}^{\circ }C$ as a function of height $h$ (in km) if the ground temperature is $30^{\circ }C$ and at $3\text{ km}$, it is $-10^{\circ }C$, assuming a linear model.
5. Given $f(x)=|x-2|$, find $f(-3)$ and sketch its graph.

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