Elementary Functions Part 1, Functions Lecture 1.1b, Functions - PowerPoint PPT Presentation

Elementary Functions Part 1, Functions Lecture 1.1b, Functions defined by equations Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 13 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Functions defined by equations Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation 2 x + 3 y − 4 = 0 define a function with inputs x and outputs y ? Isolate y to get 3 y = 4 − 2 x and so y = 1 3(4 − 2 x ) . We may now explicitly define the function f ( x ) = 1 3(4 − 2 x ) . So YES , the equation 2 x + 3 y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27

Independent and dependent variables A digression. When we considered the equation 2 x + 3 y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom !) that y is the input and x is the output. Then, solving for x , we have 2 x = 4 − 3 y and so x = 1 2(4 − 3 y ) and so we create the function g ( y ) = 1 2(4 − 3 y ) . But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. Smith (SHSU) Elementary Functions 2013 15 / 27

Independent and dependent variables A digression. When we considered the equation 2 x + 3 y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom !) that y is the input and x is the output. Then, solving for x , we have 2 x = 4 − 3 y and so x = 1 2(4 − 3 y ) and so we create the function g ( y ) = 1 2(4 − 3 y ) . But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. Smith (SHSU) Elementary Functions 2013 15 / 27

Independent and dependent variables A digression. When we considered the equation 2 x + 3 y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom !) that y is the input and x is the output. Then, solving for x , we have 2 x = 4 − 3 y and so x = 1 2(4 − 3 y ) and so we create the function g ( y ) = 1 2(4 − 3 y ) . But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. Smith (SHSU) Elementary Functions 2013 15 / 27

Independent and dependent variables A digression. When we considered the equation 2 x + 3 y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom !) that y is the input and x is the output. Then, solving for x , we have 2 x = 4 − 3 y and so x = 1 2(4 − 3 y ) and so we create the function g ( y ) = 1 2(4 − 3 y ) . But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. Smith (SHSU) Elementary Functions 2013 15 / 27

Independent and dependent variables A digression. When we considered the equation 2 x + 3 y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom !) that y is the input and x is the output. Then, solving for x , we have 2 x = 4 − 3 y and so x = 1 2(4 − 3 y ) and so we create the function g ( y ) = 1 2(4 − 3 y ) . But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. Smith (SHSU) Elementary Functions 2013 15 / 27