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Functions and Their Graphs

Explore the fundamentals of mathematical functions and their graphical representations. Learn about elementary functions such as constant, linear, quadratic, and cubic, and their distinct graphs. Understand specialized functions with restricted domains, like square root and absolute value functions. Delve into the asymptotic behavior of reciprocal functions, the growth patterns of exponential functions, and the periodic nature of trigonometric functions. Grasp the use of graphical tests like the vertical and horizontal line tests to identify function types.

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1

Function Definition in Mathematics

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A function is a relation where each input has one unique output; x maps to one y.

2

Function Graph Axes Representation

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In a function's graph, x-axis represents domain (inputs) and y-axis represents range (outputs).

3

Analyzing Function Graph Characteristics

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Graph analysis reveals continuity, slope, symmetry, indicating function's behavior.

4

A ______ function is represented by the equation f(x) = mx + b, where 'm' stands for the ______ and 'b' indicates the ______.

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linear slope y-intercept

5

The graph of a ______ function, expressed as f(x) = ax^2 + bx + c, takes the shape of a ______, which may open ______ or ______.

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quadratic parabola upwards downwards

6

Domain of Square Root Function

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f(x) = √x defined for x ≥ 0

7

Domain of Cube Root Function

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f(x) = ∛x defined for all real numbers

8

Graph Shape of Absolute Value Function

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f(x) = |x| has V-shaped graph

9

In the function f(x) = 1/x, the graph never touches but gets infinitely close to the ______ asymptote at y = 0 and the ______ asymptote at x = 0.

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horizontal vertical

10

The function f(x) = 1/x^2 only nears the x-axis from ______ due to the square in the denominator, ensuring all y values remain ______.

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above positive

11

Exponential function horizontal asymptote location

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Exponential functions have a horizontal asymptote at y = 0.

12

Logarithmic function vertical asymptote location

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Logarithmic functions have a vertical asymptote at x = 0.

13

The functions sine and cosine repeat their values every ______ radians, which is equivalent to 360 degrees.

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2π

14

Vertical Line Test Purpose

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Determines if curve is a function by checking for any vertical line intersecting curve at multiple points.

15

Horizontal Line Test Purpose

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Checks if function is injective by seeing if horizontal lines intersect graph more than once.

16

Injective Function Definition

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A function where each element of the range is mapped to by at most one element of the domain.

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Fundamentals of Functions and Their Graphical Representation

In mathematics, a function is a special relationship where each input (or element of the domain) corresponds to exactly one output (or element of the range). Typically, the independent variable is denoted by x, and the dependent variable is denoted by y. The graph of a function is a visual representation of this relationship in a coordinate system, where the horizontal axis (x-axis) represents the domain and the vertical axis (y-axis) represents the range. By analyzing a function's graph, one can discern important characteristics such as continuity, slope, and symmetry, which provide insight into the function's behavior.
Clear acrylic graphing board with etched Cartesian coordinates and a curving line, compass on blank paper, wooden ruler, and potted green plant on a desk.

Graphs of Elementary Functions and Their Features

Elementary functions include constant, linear, quadratic, and cubic functions, each with a distinctive graph. A constant function, f(x) = c, is graphed as a horizontal line that intersects the y-axis at the point (0, c). Linear functions have the form f(x) = mx + b, where m is the slope and b is the y-intercept, resulting in a straight line. Quadratic functions, f(x) = ax^2 + bx + c, create parabolas that can open upwards or downwards depending on the sign of the coefficient a. Cubic functions, f(x) = ax^3 + bx^2 + cx + d, display an S-shaped curve with inflection points and can increase or decrease without bound as x approaches positive or negative infinity.

Graphs of Specialized Functions and Domain Considerations

Certain functions are defined in such a way that they have restricted domains or unique graph shapes. The square root function, f(x) = √x, is only defined for x ≥ 0, resulting in a graph that starts at the origin and increases to the right. The cube root function, f(x) = ∛x, is defined for all real numbers, producing a graph that extends in both the positive and negative directions of the x-axis. The absolute value function, f(x) = |x|, has a graph with a V-shape, indicating that it outputs positive values for both positive and negative inputs of x.

Asymptotic Behavior and Function Graphs

Asymptotic behavior occurs when the graph of a function approaches a line arbitrarily closely but never actually reaches it. The reciprocal function, f(x) = 1/x, features two asymptotes: a horizontal asymptote along the x-axis (y = 0) and a vertical asymptote along the y-axis (x = 0). The graph of the reciprocal squared function, f(x) = 1/x^2, only approaches the x-axis from above, as the square in the denominator ensures all y values are positive, and it also has a vertical asymptote along the y-axis.

Exponential and Logarithmic Functions: Growth, Inverses, and Asymptotes

Exponential functions, exemplified by f(x) = e^x, show rapid growth or decay and have a horizontal asymptote, typically at y = 0. The graph intersects the y-axis at (0, 1) and increases (or decreases) rapidly from there. Logarithmic functions, such as f(x) = ln(x), are the inverses of exponential functions and have a vertical asymptote at x = 0. Their graphs reflect across the line y = x when compared to their exponential counterparts and pass through the point (1, 0).

Periodic Nature of Trigonometric Function Graphs

Trigonometric functions, including sine, cosine, and tangent, exhibit periodic behavior, repeating their values in regular intervals. The sine and cosine functions have a maximum amplitude of 1 and a period of 2π radians (360 degrees). The tangent function has a period of π radians and displays vertical asymptotes at odd multiples of π/2, due to its undefined values at these points.

Graphical Tests for Function Identification

The vertical line test is a graphical method used to determine if a curve represents a function; if any vertical line intersects the curve at more than one point, the curve is not the graph of a function. The horizontal line test is used to determine if a function is injective (or one-to-one); if a horizontal line intersects the graph more than once, the function is not injective. These tests are essential tools for analyzing the graphical representation of functions and understanding their characteristics.