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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A function is a special relationship where each input corresponds to exactly one output
The independent variable, denoted by x, corresponds to the input, while the dependent variable, denoted by y, corresponds to the output
The graph of a function is a visual representation of the relationship between the input and output in a coordinate system
A constant function is graphed as a horizontal line with a y-intercept at the point (0, c)
A linear function has the form f(x) = mx + b and results in a straight line on a graph
A quadratic function, f(x) = ax^2 + bx + c, creates a parabola on a graph
The square root function, f(x) = √x, is only defined for x ≥ 0 and produces 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 and produces a graph that extends in both positive and negative directions
The absolute value function, f(x) = |x|, has a V-shape on a graph and outputs positive values for both positive and negative inputs
The reciprocal function, f(x) = 1/x, has two asymptotes and approaches the x-axis from above
The reciprocal squared function, f(x) = 1/x^2, only approaches the x-axis from above and has a vertical asymptote along the y-axis
Exponential functions, such as f(x) = e^x, show rapid growth or decay and have a horizontal asymptote, while logarithmic functions, such as f(x) = ln(x), are the inverses of exponential functions and have a vertical asymptote at x = 0
Trigonometric functions, such as sine, cosine, and tangent, exhibit periodic behavior and have specific amplitudes and periods
The tangent function has vertical asymptotes at odd multiples of π/2, while the sine and cosine functions have a maximum amplitude of 1 and a period of 2π radians
The vertical line test is used to determine if a curve represents a function
The horizontal line test is used to determine if a function is injective (or one-to-one)
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Function Definition in Mathematics
A function is a relation where each input has one unique output; x maps to one y.
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Function Graph Axes Representation
In a function's graph, x-axis represents domain (inputs) and y-axis represents range (outputs).
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Analyzing Function Graph Characteristics
Graph analysis reveals continuity, slope, symmetry, indicating function's behavior.
