Lesson 3.1 Reciprocal of a Linear Function
A rational function has the form h(x) = , where f(x) and g(x) are polynomialsThe domain of a rational function consists of all real number except the zeroes of the polynomial in the denominator. g(x) ≠ 0
The zeroes of h(x) are the zeroes of f(x) if h(x) is in simplified form.
Ex 1. Consider the function f(x) = 1 / 2x – 1
a) State the domain.
b) State the equation of the vertical asymptote.
c) Describe the behavior of the function near the vertical asymptote.
d) Describe the end behavior.
e) State the equation of the horizontal asymptote.
f) Sketch the graph of the function.
g) State the range
Solution:
a) Domain = {x/ x є ___, and x ≠ ___}
b) x = _____
c)
As x → ½- :
x / f(x)-1
0.4
0.45 / -10
0.49
As x → ½+ :
x / f(x)1 / 1
0.6
0.55 / 10
0.51
d) As x → - ∞:
X / f(x)-10 / -1/21
-100
-1000 / -1/2001
-10000
As x → +∞ :
X / f(x)10 / 1/19
100
1000 / 1/1999
10000
e) y = _____
f) Sketch the graph:
Ex 2. Determine the x- and y- intercepts of the function g(x) = 2
x + 5
KEY CONCEPTS:
· The reciprocal of a linear function has the form f(x) = 1
o kx – c
· The domain: x є R, x ≠ c/k, and range: y є R, y ≠ 0
· The vertical asymptote: x = c/k, and the horizontal asymptote: y = 0
· If k> 0, the left branch of this function has a negative, decreasing slope, and the right branch has negative, and increasing slope.
· If k < 0, the left branch has positive, increasing slope, and the right branch has positive decreasing slope.