Basic Mathematics · Unit 1
Functions and Their Properties
Exam-focused notes for Functions and Their Properties (Basic Mathematics, MTH104): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Definition and notation of functions
- Domain and range of functions
- Even and odd functions
- Absolute value functions
- Function composition
- Piecewise defined functions
- Graph transformations and shifts
- Symmetry properties
Absolute value functions
Define absolute value function and Sketch the graph of absolute value. [5]
The absolute value function is defined as: $$f(x) = x = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ In other words, the absolute value of a number is its distance from zero on the number line, always expressed as a non-negat...
Full solved answer →Domain and range of functions
Find the domain and range of the function $f(x) = \sqrt{5x + 10}$. Draw the graph of the function $y = x^2$ shifted up by 1 unit, down by 2 units, also shift 3 units to left, and 2 units right with new position of function. [2+3]
- Function: $f(x) = \sqrt{5x + 10}$ - Base function to transform: $y = x^2$ - Transformations: up 1 unit, down 2 units, left 3 units, right 2 units --- (a) Domain and Range of $f(x) = \sqrt{5x + 10}$ The expression under the square root must be non-negative...
Full solved answer →Sketch the graph of $f(x) = \sqrt{x}$. Also, find the domain and range. [5]
The function f(x) = √x is defined only for non-negative real numbers (since we cannot take the square root of negative numbers in the real number system). Domain: [0, ∞) or {x ∈ ℝ x ≥ 0} Since √x produces only non-negative outputs: - When x = 0, f(0) = 0 - ...
Full solved answer →Symmetry properties
Graph the following functions. Write their symmetricity and specify the interval over which the function is increasing and decreasing. $y = -x^3$, $y = x^2$. [5]
- Function 1: $y = -x^3$ - Function 2: $y = x^2$ - Required: graph, symmetry, intervals of increase/decrease. --- $x$ $-2$ $-1$ $0$ $1$ $2$ ------------------ $y=-x^3$ $8$ $1$ $0$ $-1$ $-8$ A cubic curve passing through the origin, rising steeply from the u...
Full solved answer →Even and odd functions
Even and Odd Functions
Definition: A function $f(x)$ is even if $f(-x) = f(x)$ for every $x$ in its domain. Symmetry: The graph is symmetric about the y-axis. Example: $f(x) = x^2$ $$f(-x) = (-x)^2 = x^2 = f(x) \checkmark$$ Definition: A function $f(x)$ is odd if $f(-x) = -f(x)$ ...
Full solved answer →Question
In 2000, $100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5.5%. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount $A$ in the account after $x$ years have elapsed.
Define when the function $f(x)$ is odd and even. Also, define when a function $f(x)$ is increasing and decreasing? If $y = x^2$ is a given function then determine the interval in which the function is increasing and decreasing and draw the graph of the given function. [5+5]
- Principal (initial investment in year 2000): $P = 100$ - Annual interest rate: $r = 5.5\% = 0.055$ - Compounding frequency: once per year (annually) - No additional deposits or withdrawals - Time variable: $x$ = number of years elapsed since 2000 - Functi...
Full solved answer →Make Unit 1 stick
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