Understanding And Working With Functions
From Zero to Hero: Becoming a Function Expert
Start
Index
Linear Functions
Polynomial Functions
Logarithmic Functions
Inverse Functions
Domain
Range
Zeroes
Extrema
Asymptote
A linear function is a function in which the relationship between the two variables is a straight line.
The equation y = 2x + 1 is a linear function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a straight line.
Another example of a linear function is the equation y = -4x + 3, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a straight line that has a negative slope.
A polynomial function is a function in which the relationship between the two variables is a polynomial expression.
The equation y = x^2 + 3x + 4 is a polynomial function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a parabola.
Another example of a polynomial function is the equation y = x^3 + 2x^2 + 5x + 7, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a cubic curve.
A logarithmic function is a function in which the relationship between the two variables is an exponential expression.
The equation y = log2x is a logarithmic function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that increases exponentially.
Another example of a logarithmic function is the equation y = log3x, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that increases exponentially at a faster rate than the previous example.
An inverse function is a function in which the relationship between the two variables is reversed, such that the independent variable becomes the dependent variable and vice versa.
The equation y = 3/x is an inverse function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that decreases exponentially.
A second example of an inverse function is the equation y = 2/x, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that decreases exponentially at a faster rate than the previous example.
The domain of a function is the set of all possible values of the independent variable.
The equation y = 2x + 1 has a domain of all real numbers, as any real number can be used for the independent variable x.
Equation y = 2/x has a domain of all non-zero real numbers, as any non-zero real number can be used for the independent variable x.
The range of a function is the set of all possible values of the dependent variable.
Equation y = 2x + 1 has a range of all real numbers, as any real number can be the result of the dependent variable y.
The equation y = 2/x has a range of all positive real numbers, as any positive real number can be the result of the dependent variable y.
Zeroes of a function are the points where the dependent variable is equal to zero.
Equation y = x^2+8x+12 has two zeroes, at x = -6 and x = -2. At both of these points, the dependent variable y is equal to zero.
The equation y = 3x^2 + 8x - 12 has three zeroes, at x = -4, x = 0, and x = 3. At each of these points, the dependent variable y is equal to zero.
Extrema of a function are the points where the dependent variable is at its maximum or minimum value.
Equation y = x^2 - 6x + 9 has two extrema, at x = 3 and x = 9. At x = 3, the dependent variable y is at its minimum value, and at x = 9 it is at its maximum value.
The equation y = x^3 - 9x + 4 has one extrema, at x = 3. At this point, the dependent variable y is at its maximum value.
An asymptote of a function is a line that the function approaches, but never crosses or touches.
The equation y = 1/x has an asymptote at x = 0. As the independent variable x approaches zero, the dependent variable y approaches infinity, but never actually reaches zero or infinity.
The equation y = 1/x^2 has two asymptotes, at x = 0 and x = infinity. As the independent variable x approaches zero from the left, the dependent variable y approaches infinity, and as the independent variable x approaches infinity from the right, the dependent variable y approaches zero.
Summary
A function is a mathematical relationship between two variables, where the value of one variable (the dependent variable) is determined by the value of another variable (the independent variable).
What is a function?
Terry Presswood
Created on October 27, 2023
Basic definition list for functions and types
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Transcript
Understanding And Working With Functions
From Zero to Hero: Becoming a Function Expert
Start
Index
Linear Functions
Polynomial Functions
Logarithmic Functions
Inverse Functions
Domain
Range
Zeroes
Extrema
Asymptote
A linear function is a function in which the relationship between the two variables is a straight line.
The equation y = 2x + 1 is a linear function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a straight line.
Another example of a linear function is the equation y = -4x + 3, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a straight line that has a negative slope.
A polynomial function is a function in which the relationship between the two variables is a polynomial expression.
The equation y = x^2 + 3x + 4 is a polynomial function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a parabola.
Another example of a polynomial function is the equation y = x^3 + 2x^2 + 5x + 7, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a cubic curve.
A logarithmic function is a function in which the relationship between the two variables is an exponential expression.
The equation y = log2x is a logarithmic function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that increases exponentially.
Another example of a logarithmic function is the equation y = log3x, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that increases exponentially at a faster rate than the previous example.
An inverse function is a function in which the relationship between the two variables is reversed, such that the independent variable becomes the dependent variable and vice versa.
The equation y = 3/x is an inverse function, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that decreases exponentially.
A second example of an inverse function is the equation y = 2/x, with x being the independent variable and y being the dependent variable. When plotting this equation on a graph, the result will be a curve that decreases exponentially at a faster rate than the previous example.
The domain of a function is the set of all possible values of the independent variable.
The equation y = 2x + 1 has a domain of all real numbers, as any real number can be used for the independent variable x.
Equation y = 2/x has a domain of all non-zero real numbers, as any non-zero real number can be used for the independent variable x.
The range of a function is the set of all possible values of the dependent variable.
Equation y = 2x + 1 has a range of all real numbers, as any real number can be the result of the dependent variable y.
The equation y = 2/x has a range of all positive real numbers, as any positive real number can be the result of the dependent variable y.
Zeroes of a function are the points where the dependent variable is equal to zero.
Equation y = x^2+8x+12 has two zeroes, at x = -6 and x = -2. At both of these points, the dependent variable y is equal to zero.
The equation y = 3x^2 + 8x - 12 has three zeroes, at x = -4, x = 0, and x = 3. At each of these points, the dependent variable y is equal to zero.
Extrema of a function are the points where the dependent variable is at its maximum or minimum value.
Equation y = x^2 - 6x + 9 has two extrema, at x = 3 and x = 9. At x = 3, the dependent variable y is at its minimum value, and at x = 9 it is at its maximum value.
The equation y = x^3 - 9x + 4 has one extrema, at x = 3. At this point, the dependent variable y is at its maximum value.
An asymptote of a function is a line that the function approaches, but never crosses or touches.
The equation y = 1/x has an asymptote at x = 0. As the independent variable x approaches zero, the dependent variable y approaches infinity, but never actually reaches zero or infinity.
The equation y = 1/x^2 has two asymptotes, at x = 0 and x = infinity. As the independent variable x approaches zero from the left, the dependent variable y approaches infinity, and as the independent variable x approaches infinity from the right, the dependent variable y approaches zero.
Summary
A function is a mathematical relationship between two variables, where the value of one variable (the dependent variable) is determined by the value of another variable (the independent variable).