Functions

Definition and Properties of Functions

In calculus, a function is a rule that assigns a single output value to each input value. Functions are fundamental objects in mathematics, especially in calculus, where they are used to describe relationships between varying quantities.

• Notation: If the input is x and the function is f, the output is written as f(x).

• Graph: The graph of a function f is the set of all points (x, y) in the Cartesian plane such that y = f(x).

• Independent and Dependent Variables: The input variable is called the independent variable, and the output is the dependent variable.

• Domain Restrictions: Some functions are only defined for certain input values.

• Piecewise Functions: Some functions are defined by different formulas for different input values.

• Function Notation: It is customary to refer to f(x) rather than just f to avoid ambiguity.

Example: If f(x) = x^2 - x + 1, then f(2) = 2^2 - 2 + 1 = 3.

Inverse Functions

An inverse function reverses the effect of the original function, mapping outputs back to their corresponding inputs. Not all functions have inverses; a function must be one-to-one (injective) for its inverse to exist.

• Notation: The inverse of f is denoted f^{-1}.

• Example: The exponential function a^x and the logarithmic function \log_a x are inverses: if y = a^x, then x = \log_a y.

• Restriction for Inverses: If a function is not one-to-one, its domain may be restricted to ensure the existence of an inverse. For example, \sqrt{x} is the inverse of x^2 restricted to x \geq 0.

Example: The inverse of f(x) = x^2 (for x \geq 0) is f^{-1}(x) = \sqrt{x}.

Radian Measure

Definition and Conversion

Angles can be measured in radians, a natural unit based on the arc length of a circle. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius.

• Arc Length Formula:

• Conversion: , where \alpha is in degrees and \theta is in radians.

• Key Values:

  • 360° = 2\pi radians

  • 180° = \pi radians

  • 90° = \frac{\pi}{2} radians

  • 60° = \frac{\pi}{3} radians

  • 45° = \frac{\pi}{4} radians

  • 30° = \frac{\pi}{6} radians

Example: Convert 60° to radians: radians.

Area of a Sector

The area of a sector of a circle with radius r and angle \theta (in radians) is given by:

• Formula:

Example: For a sector with r = 4 and \theta = \frac{1}{2} radians, .

Applications: Goat in a Square Field

Consider a goat tied to the corner of a square field of side 100 m with a rope of length r. The goat can graze an area equal to a quarter circle of radius r plus a region within the square. If the goat is to reach exactly half the grass in the field, we set the grazed area equal to half the area of the square.

• Area of Square: m2

• Area Grazed:

• Set Equal:

• Solve for r:

Additional info: This classic problem demonstrates the use of radian measure and area of a sector in a real-world context.

Trigonometric Functions

Definition and Geometric Interpretation

Trigonometric functions relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, they are defined as follows for an angle \theta:

• cosine:

• sine:

• tangent:

On the unit circle (r = 1), the coordinates of a point are (\cos \theta, \sin \theta).

Other Trigonometric Functions

• cotangent:

• secant:

• cosecant:

Key Identities and Properties

• Pythagorean Identity:

• Other Identities:

• Angle Addition:

• Double Angle:

• Even-Odd Properties:

Solving Trigonometric Equations

Trigonometric equations can be solved by expressing them in terms of sine, cosine, or tangent and using the identities above. The general solutions are:

• or

Example: Solve . Use identities to rewrite and solve for \theta.

Standard Angles and Triangles

Values of trigonometric functions for standard angles (such as , , ) are derived from special right triangles and are frequently used in calculus.

Summary Table: Trigonometric Function Values for Standard Angles

Additional info: Mastery of these values and identities is essential for calculus, especially for differentiation and integration involving trigonometric functions.