Calculus 1 - Lecture 1 Notes

Functions: Definition and Notation

• A function is a rule that assigns each input value x to a unique output value y.
• The set of all possible input values x is called the domain.
• The set of all corresponding output values y (also denoted as f(x)) is called the range.
• It is crucial that each x value is associated with only one unique y value.

Visualizing Functions

• Think of a function as a process: input goes in, the function rule is applied, and an output comes out.
• Domain values are the inputs, and range values are the outputs.
• Each x in the domain must have its own unique y value in the range.
• Multiple x values can map to the same y value, but one x cannot map to multiple y values.

Function Notation

• f(x) = x^2 - 2x: This is a function rule where f(x) represents the output when x is the input.
• f(-1): This means we substitute -1 for every instance of x in the function rule.
  • f(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3

Evaluating Functions with Expressions

• We can substitute entire expressions for x as well:
  • f(x^2) = (x^2)^2 - 2(x^2) = x^4 - 2x^2

Graphical Representation of Functions

• If a graph has two points with the same x value but different y values, it is not a function.
  • This is because one x value is associated with two different y values.
• Having points with the same y value but different x values is acceptable for a function.

Vertical Line Test

• The vertical line test is a visual method to determine if a graph represents a function.
• If any vertical line intersects the graph more than once, the graph does not represent a function.
• A parabola is an example of a function because no vertical line will intersect it more than once.
• A circle is not a function because a vertical line can intersect it at two points.

Domain and Range from a Graph

• Domain: all possible x values.
• Range: all possible y values.
• For a parabola that extends infinitely in both horizontal directions, the domain is all real numbers (-\infty, \infty).
• If the parabola's lowest point is at y = 1, then the range is all y values greater than or equal to 1, which is [1, \infty).

Domain and Range: Algebraic Approach

• For y = x^2 + 1, we can input any real number for x. Thus, the domain is all real numbers.
• Since x^2 is always non-negative, the smallest value of y is 1 (when x = 0). Thus, the range is [1, \infty).

Example: f(x) = \sqrt{4 - x^2}

• Domain: the set of all x values for which the function is defined in the real number system.
• Range: The set of output values given the domain.

Determining Domain Algebraically

• Since we cannot take the square root of a negative number in the real number system, the expression inside the square root must be greater than or equal to zero.
  • 4 - x^2 \geq 0
  • -x^2 \geq -4
  • x^2 \leq 4
• This inequality holds true for x values between -2 and 2, inclusive.
  • [-2, 2]

Determining Range Algebraically

• The smallest value inside the square root is 0 (when x = 2 or x = -2), resulting in \sqrt{0} = 0.
• The largest value inside the square root is 4 (when x = 0), resulting in \sqrt{4} = 2.
• Therefore, the range is [0, 2].

Verification

• When in doubt, use a graphing calculator to visualize the function and confirm the domain and range found algebraically.