Lines and Functions
Lines and Slope
Recall that the slope of a line through points (x1,y1) and (x2,y2) is the rise over the run or
y2 - y1
For a function f(x) the secant line between x = a and x = b is the line through the points (a,f(a)) and (b,f(b))
Example Finding the slope of the secant line
Find the equation of the secant line for y = x2 between x = 1 and x = 3.
At x = 1,
f(1) = 12 = 1
and at x = 2,
f(2) = 22 = 4
We need the equation of the line through the points (1,1) and (2,4). The slope of the line is
4 - 1
If we know that the slope of a line is m and the line passes through the point (x1,y1) then the equation of the line can be found by the using the formula:
y - y1 = m(x - x1)
y - 1 = 3(x - 1) = 3x - 3
y = 3x - 2 adding 1 to both sides
Two lines can be parallel , perpendicular, or neither.
If they are parallel, then they have the same slope: m1 = m2
they are perpendicular if the slopes are negative reciprocals of each other: m1 = -1/m2
y = 3x - 4 and y = 3x + 7
are parallel since they have the same slope.
and the lines
y = -5x + 2 and y = 1/5 x - 1
are perpendicular since if we multiply the slopes together, the product is -1.
(-5)(1/5) = -1
Recall that a function is a rule that assigns to each element of one set called the domain a unique element of another set called the range.
Blood Pressure, the domain is the set of people and the range is the set of possible blood pressures.
Composition of Functions
f(x) = 3x + 1 and g(x) = 2x - 1
f(x + 2) - f(x)
A. f(g(x)) = 3(2x - 1) + 1 = 6x - 2
B. First notice that
f(x + 2) = 3(x + 2) + 1
f(x + 2) - f(x) = 3(x + 2) + 1 - (3x + 1) = 3x + 6 + 1 - 3x - 1
With g as in the example above find
g(x + 3) - g(x)
Exercise: Suppose that
f(x) = Pekt, f(0) = 10, f(1) = 25
Inverses: If f passes the horizontal line test then f has an inverse.
To find an inverse we
1) switch x and y
2) solve for y
Find the inverse of
x + 1
We begin by switching the x's and y's.
y + 1
x(2y - 5) = y + 1 Multiply by 2y - 5
2xy - 5x = y + 1 Distribute the x through
2xy - y = 5x + 1 Bring the y terms to the left and the others to the right
y(2x - 1) = 5x + 1 Factor out a y
5x + 1
We can conclude that
5x + 1