Types of Functions in Mathematics

Types of Functions

Linear Functions:
- Form: f(x) = mx + b
- Characteristics: Straight line; constant rate of change.
Quadratic Functions:
- Form: f(x) = ax² + bx + c
- Characteristics: Parabolic shape; vertex and axis of symmetry.
Polynomial Functions:
- Form: f(x) = aₙxⁿ + ... + a₁x + a₀
- Characteristics: Smooth curves; degree determines end behavior.
Rational Functions:
- Form: f(x) = P(x)/Q(x), Q(x) ≠ 0
- Characteristics: May have asymptotes; discontinuities at zeros of Q(x).
Exponential Functions:
- Form: f(x) = a * b^x
- Characteristics: Rapid growth or decay; constant multiplicative rate.
Logarithmic Functions:
- Form: f(x) = log_b(x)
- Characteristics: Inverse of exponential functions; slow growth.
Trigonometric Functions:
- Includes sine, cosine, tangent, etc.
- Characteristics: Periodic behavior; defined on unit circle.

Properties of Functions

Domain: Set of possible input values (x-values).
Range: Set of possible output values (y-values).
Continuity: Function is continuous if it can be drawn without lifting pencil.
Boundedness: Function is bounded if it has a maximum/minimum value.
Even Functions: f(-x) = f(x); symmetric about the y-axis.
Odd Functions: f(-x) = -f(x); symmetric about the origin.
One-to-One: Each output is produced by exactly one input.

Inverse Functions

Definition: If f(x) maps x to y, then f⁻¹(y) maps y back to x.
Finding Inverses:
1. Replace f(x) with y.
2. Solve for x in terms of y.
3. Swap x and y.
Graphical Interpretation: The graph of f and f⁻¹ are reflections over the line y = x.
Conditions: A function must be one-to-one to have an inverse.

Function Transformations

Vertical Shifts: f(x) ± k (moves up/down).
Horizontal Shifts: f(x ± h) (moves left/right).
Vertical Stretch/Compression: k * f(x) (k > 1 stretches, 0 < k < 1 compresses).
Horizontal Stretch/Compression: f(kx) (k > 1 compresses, 0 < k < 1 stretches).
Reflections: f(-x) (reflects over y-axis); -f(x) (reflects over x-axis).

Graphing Techniques

Intercepts: Finding where f(x) = 0 (x-intercepts) and f(0) (y-intercept).
Asymptotes: Lines that the graph approaches but never touches (vertical and horizontal).
Testing Intervals: Use test points to determine where the function is positive or negative.
End Behavior: Analyze limits as x approaches ±∞ to predict graph behavior at extremes.
Critical Points: Identify points where the derivative equals zero or is undefined; helps locate maxima and minima.

Types of Functions

Linear Functions
- Represented by the equation f(x) = mx + b
- Graph is a straight line
- The slope (m) represents the rate of change and the y-intercept (b) is the point where the line crosses the y-axis.
Quadratic Functions
- Represented by the equation f(x) = ax² + bx + c
- Graph is a parabola
- The vertex is the highest or lowest point on the parabola, depending on the sign of the coefficient 'a'.
- Parabolas have an axis of symmetry, which is a vertical line that passes through the vertex.
Polynomial Functions
- General form: f(x) = aₙxⁿ +...+ a₁x + a₀
- Graphs are smooth curves. The degree of the polynomial (highest power of 'x') determines the end behavior of the graph - whether it rises or falls as x approaches positive or negative infinity.
Rational Functions
- General form: f(x) = P(x)/Q(x), Q(x) ≠ 0
- Graphs may have vertical asymptotes at the values of x that make Q(x) equal to zero, and horizontal asymptotes determined by the degree of the polynomials P(x) and Q(x)
- They can also have discontinuities, or holes, at values of x where both P(x) and Q(x) are zero.
Exponential Functions
- General form: f(x) = a * b^x
- Exponental functions exhibit rapid growth or decay
- The constant 'b' represents the growth or decay factor. If 'b' is greater than 1, the function grows exponentially; if 'b' is between 0 and 1, the function decays exponentially.
Logarithmic Functions
- General form: f(x) = log_b(x)
- Logarithmic functions are the inverse of exponential functions
- They grow very slowly as x increases
Trigonometric Functions
- Include functions such as sine, cosine, tangent, cotangent, secant, and cosecant
- Defined by their relationship to the unit circle: Sine is the vertical component of a point on the unit circle at a given angle, cosine is the horizontal component, and tangent is the ratio of sine to cosine.
- Characterized by their periodic behaviour - they repeat at regular intervals.

Properties of Functions

Domain - The set of all possible input values (x-values) for which the function is defined.
Range - The set of all possible output values (y-values) that the function can produce.
Continuity - A function is considered continuous if it can be graphed without lifting your pen from the paper, or if it has no gaps or jumps in its graph.
Boundedness - A function is bounded if its output values are restricted, meaning they have a maximum and/or a minimum value.
Even Functions: f(-x) = f(x) for all x in the domain - The graph of an even function is symmetric about the y-axis.
Odd Functions: f(-x) = -f(x) for all x in the domain - The graph of an odd function is symmetric about the origin.
One-to-One: A function is one-to-one if each output value is produced by exactly one input value - It passes the horizontal line test - no horizontal line intersects its graph more than once.

Inverse Functions

Definition - The inverse of a function 'f' denoted by f⁻¹, reverses the action of the original function
- If f(a) = b, then f⁻¹(b) = a
Finding Inverses
1. Replace f(x) with 'y'.
2. Solve for 'x' in terms of 'y'.
3. Swap 'x' and 'y'.
Graphical Interpretation - The graphs of a function and its inverse are reflections of each other across the line y = x
Conditions - A function must be one-to-one (passes the horizontal line test) to have an inverse function.

Function Transformations

Vertical Shifts: f(x) ± k
- Shifts the graph up by 'k' units if 'k' is positive and down by 'k' units if 'k' is negative.
Horizontal Shifts: f(x ± h)
- Shifts the graph left by 'h' units if 'h' is positive and right by 'h' units if 'h' is negative.
Vertical Stretch/Compression: k * f(x)
- Stretches the graph vertically by a factor of 'k' if 'k' is greater than 1
- Compresses the graph vertically by a factor of 'k' if 'k' is between 0 and 1
Horizontal Stretch/Compression: f(kx)
- Compresses the graph horizontally by a factor of 'k' if 'k' is greater than 1
- Stretches the graph horizontally by a factor of 'k' if 'k' is between 0 and 1
Reflections:
- f(-x) reflects the graph over the y-axis.
- -f(x) reflects the graph over the x-axis.

Graphing Techniques

Intercepts:
- x-intercepts: occur where f(x) = 0 - these are the points where the graph crosses the x-axis.
- y-intercept: occurs where x = 0 (f(0)) - this is the point where the graph crosses the y-axis.
Asymptotes - Lines that the graph approaches but never touches
- Vertical asymptotes occur at values of 'x' that make the function undefined:
  - Often happen at x-values that make the denominator of a rational function equal to zero.
- Horizontal asymptotes are lines that the graph approaches as x approaches positive or negative infinity.
Testing Intervals: Use test points in each interval where f(x) is either positive or negative to determine the overall shape of the graph.
End Behavior: Analyze the limits of f(x) as x approaches positive or negative infinity to determine where the graph goes as it extends toward the extremes of the x-axis.
Critical Points: Points where the derivative of the function is equal to zero or undefined.
- These points are important because they often correspond to the locations of relative maxima and minima (peaks and valleys) on the curve.