Functions II Lecture Notes

Functions II Lecture Notes

Dr. Emmanuel Akweittey
Department of Mathematics
Kwame Nkrumah University of Science and Technology
KNUST ranks No.1 globally for Quality Education (SDG4)

Dr. Gabby | Math KNUST | gabriel.of@knust.edu.gh

Lecture Outline

1. Properties of Functions
   • Odd and Even Functions
   • Periodic Functions
   • Monotonic Functions
   • Bounded Functions
   • Maxima and Minima of Functions
2. Inverse Function

3. Sequence and Series

Properties of Functions

Odd and Even Functions

Even Function

• Let $f$ be a function and $Df$ its domain.
• Assumption: If $x \in Df$ then $-x \in Df$.
• Definition (Even Function):
  • A function $f$ is an even function if
    $f(-x) = f(x).$
• Examples of Even Functions:
  • $f(x) = x^2$
  • Check: $f(-x) = (-x)^2 = x^2 = f(x)$
  • $g(x) = -x^4 + 2x^2 - 1$
  • Check: $g(-x) = -(-x)^4 + 2(-x)^2 - 1 = g(x)$
  • $h(x) = ext{cos}(x) + x^2$
  • Check: $h(-x) = ext{cos}(-x) + (-x)^2 = h(x)$
  • $i(x) = x ext{sin}(x)$

- Check: $i(-x) = -x(- ext{sin}(x)) = x ext{sin}(x) = i(x)$

Odd Function

• Definition (Odd Function):
  • A function $f$ is an odd function if
    $f(-x) = -f(x).$
• Examples of Odd Functions:
  • $f(x) = x$
  • Check: $f(-x) = -x = -f(x)$
  • $f(x) = -x^3 + 2x$
  • Check: $f(-x) = -(-x)^3 + 2(-x) = x^3 - 2x = -f(x)$
  • $f(x) = ext{sin}(x)$
  • Check: $f(-x) = ext{sin}(-x) = - ext{sin}(x) = -f(x)$
  • $f(x) = ext{csc}(x)$
  • Check: $f(-x) = - ext{csc}(x) = -f(x)$
  • $f(x) = ext{tan}(x)$

- Check: $f(-x) = - ext{tan}(x) = -f(x)$

Remarks:

• The graph of an even function is symmetric about the y-axis.

- The graph of an odd function is symmetric about the origin.

Periodic Functions

Definition:

• Let $f$ be a function and $Df$ its domain. A function $f$ is a periodic function if there exists a positive real number $t$ such that
  $f(x+t) = f(x) ext{ for all } x \in Df.$
• The minimum of such $t$'s is denoted as $T$, called the period of $f$.
• Examples of Periodic Functions:
  • Trigonometric functions
  • $ext{sin}(x + 2k ext{π}) = ext{sin}(x) ext{ for } k \in ext{Z}, T = 2 ext{π}.$
  • $ext{cos}(x + 2k ext{π}) = ext{cos}(x), T = 2 ext{π}.$

- $ext{tan}(x + (2k + 1) ext{π}) = ext{tan}(x + ext{π}), T = ext{π}.$

Monotonic Functions

Definition:

• Let $I$ be an open interval, $x1$ and $x2$ are two elements of $I$ such that $x1 < x2$.
• Function behaviors:
  • $f$ is an increasing function on $I$ if
    $f(x<em>1) < f(x</em>2).$
  • $f$ is a decreasing function on $I$ if
    $f(x<em>1) > f(x</em>2).$
• Examples of Increasing Functions:
  • $e^x$
  • $ ext{tan}(x)$
  • $ax + b$, where $a > 0$
• Examples of Decreasing Functions:
  • $e^{-x}$
  • $ ext{cot}(x)$

- $ax + b$, where $a < 0$

Monotonic Function Examples

Monotonic Decreasing Function Graphs:

- Representation of graph for functions like $-3x + 1$ and $e^{-x}$ trending downwards.

Monotonic Increasing Function Graphs:

- Representation of graph for functions like $2x - 1$ and $e^x$ trending upwards.

Mixed Monotonicity

- Functions that increase in some sections and decrease in others, e.g., $ ext{sin}(x)$, depicted in graphs.

Example: Show that $f(x) = ext{√}(x - 2)$ is an increasing function

1. Domain: $Df = [2, + ext{∞})$.
2. For $x1, x2 \in Df,$ with $x1 < x2$, following results lead to:
   • $2 < x1 < x2 \Rightarrow 0 < x1 - 2 < x2 - 2$
   • This implies:
     0 < f(x1) < f(x2).

3. Hence, $f$ is increasing on its domain.

Bounded Functions

Definition:

• A function is said to be bounded above if there exists $ar{u} \in ext{R}$ such that
  $f(x) \leq \bar{u} ext{ for all } x ext{ in the domain of } f.$
• Example: $f(x) = x^2 + 1$ defined on $0 \leq x \leq 1$ is bounded above by 2.
• Bounded functions can also be defined below.

Definition:

• A function $f$ is said to be bounded below if there exists $ar{ ext{l}} \in ext{R}$ such that
  $f(x) \geq \bar{ ext{l}} ext{ for all } x ext{ in the domain of } f.$

- Example: $g(x) = |- ext{√}(x + 1)|$ is bounded below by 0 on the interval $[0, 4]$.

Maxima and Minima of Functions

Local (Relative) and Global (Absolute) Minimum

1. A function $f$ has a local minimum value at the point $x0$ if $f(x</em>0) \leq f(x) ext{ for all } x ext{ in a neighborhood of } x_0.$
2. A global minimum value occurs at $x0$ if $f(x</em>0) \leq f(x) ext{ for all } x ext{ in the domain of } f.$
   • In this case, $f$ is bounded below.

Local and Global Maximum

1. A function $f$ has a local maximum value at $x0$ if $f(x) \leq f(x</em>0) ext{ for all } x ext{ in a neighborhood of } x_0.$
2. If a maximum is global, it occurs when
   $f(x) \leq f(x_0) ext{ for all } x ext{ in the domain of } f.$

- In this case, $f$ is bounded above.

Inverse Functions

Definition:

1. An inverse function is a function that undoes the action of another function.
2. A function $g$ is the inverse of a function $f$ if for every $y = f(x)$, $x = g(y)$.
3. The composition of the functions must satisfy:
   $g(f(x)) = x.$

4. A function $f$ has an inverse function only if each $y$ in its range corresponds to exactly one $x$ in its domain for which $f(x) = y$; the inverse function is unique, commonly denoted as $f^{-1}$.

Finding the Inverse Function

1. Given the function $f(x)$, replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$. This is the critical step.
4. Rename $y$ as $f^{-1}(x)$.

5. Verify: $(f \circ f^{-1})(x) = x$ and $(f^{-1} \circ f)(x) = x$ ought to hold true.

Example

1. For $f(x) = 3x - 2$, follow the steps to find $f^{-1}(x)$.
   • Step 1: $y = 3x - 2$
   • Step 2: Switch to $x = 3y - 2$
   • Step 3: Solve for $y$: $y = rac{1}{3}(x + 2)$

- Step 4: Thus, $f^{-1}(x) = \frac{x}{3} + \frac{2}{3}.$

Verification of Inverse

1. Check composition:

- $(f \circ f^{-1})(x) = 3\frac{x + 2}{3} - 2 = x.$

Graph of Function and Its Inverse

- Graphical representation shows the inverse of a function is a reflection across the line $y = x$.

Sequence and Series

Definitions:

• Sequence: An ordered set of numbers following a specific rule to determine subsequent terms.
• Example of a Sequence: $x, x^2, x^3, x^4,..$

- Series: A summation of terms from a sequence, represented using the Greek letter sigma, $A3$.

Properties of Series

1. Finite Series:
   • A summation of a finite number of terms.
2. Infinite Series:

- An extension to an infinite number of terms.

Theorems of Finite Series

1. A3_{i=1}^n 1 = n.
2. A3_{i=1}^n c = nc.
3. A3_{i=1}^n i = \frac{n(n + 1)}{2}.
4. A3_{i=1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}.

5. A3_{i=1}^n i^3 = \left(\frac{n(n + 1)}{2}\right)^2.

Types of Sequences

1. Arithmetic Sequence:
   • Terms differ by a constant additive value.
   • Example: {3, 6, 9, 12,…} where difference $d = 3$.
2. Geometric Sequence:
   • Terms differ by a constant multiplicative value.

- Example: {3, 9, 27, 81,…} where ratio $r = 3$.

Theorem for Arithmetic and Geometric Series

1. For arithmetic series:
   • $S_n = n \frac{(2a + (n - 1)d)}{2}.$
2. For geometric series:

- $S_n = \frac{a(r^n - 1)}{r - 1}.$

Example: Solve the Series

1. A3{i=1}^{20} (2i^2 + 7i) = A3{i=1}^{20} 2i^2 + A3_{i=1}^{20} 7i.
   • Calculate each summation using previously defined series methods:

- Find answers stepwise.

Final Tasks

1. Exercises:

- Find the period of functions, domain, and classify the given functions as odd, even, or neither.

END OF LECTURE

KNUST ranks No.1 globally for Quality Education (SDG4)

Dr. Gabby | Math KNUST | gabriel.of@knust.edu.gh