Functions
Consider the equation: y = 2x + 3, some ordered pairs that satisfy this equation are: (-2,-1), (-1,1), (0,3).
Dependent and Independent Variables :
In this equation: y = 2x +3, the value of y depends on the value of x, then “y” is the dependent variable and “x” is the independent variable.
Relation, Domain, and Range:
A relation is any set of ordered pairs of the form (x, y). The of x-coordinates is called the domain of the relation. The set of y-coordinates is called the range of the relation
Function:
Is a relation in which each element of the domain corresponds to exactly one element in the range.
Example: Give the domain and range, then determine whether the relation is a function.
A. {(1,4), (2,3), (3,5), (-1,3), (0,6)}
B. {(-1,3), (4,2), (3,1), (2,6), (3,5)}
Solution:
A. The domain is {1,2,3,-1,0} and the range is {4,3,5,6}. The number 3 was included only once, even though appears in both pairs (2,3) and (-1,3). Since no x-value corresponds to more than one y-value in the range, this relation is a function.
B. The domain is {-1,4,3,2} and the range is {3,2,1,6,5}. Since the ordered pairs (3,1) and (3,5) have the same first coordinate (x) and a different second coordinate, each value does not correspond to exactly one value in the range; therefore, this relation is not a function.
From this example, an alternate definition of function is a set of ordered pairs in which no first coordinate is repeated.
Composite Function
Given the function: f[g(x)] = (f o g)(x)
Function f[g(x)] is called a composition of f with g or the composite function of f with g
Example:
Given f(x) = x – 3 and g(x) = x + 7, find:
a) (f o g)(x)
b) (f o g)(2)
c) (g o f)(x)
d) (g o f)(2)
Solution:
a) (f o g)(x)
since f(x) = x + 4, then (f o g)(x) = f[g(x)] = (x + 7) – 3 = x + 4
(f o g)(x) = x + 4
b) (f o g)(2) = x + 4 = 2 + 4 = 6
(f o g)(2) = 6
c) (g o f)(x)
since g(x) = x + 7, then (g o f)(x) = g[f(x)] = (x – 3) + 7 = x + 4
(g o f)(x) = x + 4
d) (g o f)(2) = x + 4 = 2 +4 = 6
(g o f)(2) = 6
Inverse Function
One-to-one functions consist of ordered pairs in which each x-coordinate corresponds to exactly one y-coordinate and each y-coordinate corresponds to exactly one x-coordinate. Such a relationship allows creating new functions called the inverse function. Only one-to-one functions have inverses.
Examples:
Function f(x): {(0,1), (1,4), (-3,-2)}
The inverse function will be: f−1(x)= {(1,0), (4,1), (-2,-3)
Function f(x) = 4x + 2, this function also can be expressed as:
y = 4x + 2
Interchange x and y:
x = 4y +2, solving for y:
x – 2 = 4y
x−24=y
The inverse will be: f−1(x) = y
x−24=y
Linear Function or Linear Equation
A linear function is a function of the form f(x) = mx + b or y = mx + b
· The graph of any linear equation is a straight line.
· The domain of any linear function is all real numbers.
· If m ≠ 0, then the range of any linear function is all real numbers.
· m, is the slope of the line (is a measure of the steepness of a line)
· b, is the y-intercept
Linear Equations (mathsisfun.com)Links to an external site.
Example:
Find the linear equation that passes through the points (2,3) and (1,4) steps: 1) find the slope (m), m = y2−y1x2−x1
let (2, 3) be (x1,y1) and (1, 4) be (x2,y2)
m=y2−y1x2−x1=4−31−2=1−1=−1
2) find the point-slope form of the equation of the line
y–y1=m(x–x1), replacing the values of m, x1 , and y1 [ordered pair (2,3)]
y – 3 = – 1 (x – 2)
y – 3 = -x + 2
y = -x + 2 + 3
y = – x + 5
References
Angel, A., & Rundle, D. (2018). Intermediate algebra for college students. (10th edition). Pearson.