An asymptote is a straight line which a functions approaches but never intersects. Is that the case here?
C
c
Factor the expression in the numerator.
A
a
Graph:
Type of Function: Linear, see solution.
B
b
Relationship: Linear
f(0.9)=5.9
f(1.1)=6.1 Graph:
Is there an asymptote at x=1? No, see solution.
C
c
f(x)=x+5
Practice makes perfect
a
Let's first calculate the function's values for the given values of x.
x
x^2+4x-5/x-1
f(x)
- 2
( - 2)^2+4( - 2)-5/- 2-1
3
- 1
( - 1)^2+4( - 1)-5/- 1-1
4
0
( 0)^2+4( 0)-5/0-1
5
1
( 1)^2+4( 1)-5/1-1
Undefined
2
( 2)^2+4( 2)-5/2-1
7
3
( 3)^2+4( 3)-5/3-1
8
Now we can complete the table.
c|c|c|c|c|c|c
x & - 2 & - 1 & 0 & 1 & 2 & 3
y & 3 & 4 & 5 & undefined & 7 & 8
Finally, we will graph the function. Notice that the function is not defined for x=1, which we can show as an open point.
b
From Part A, we see that the function is a straight line which means there is a linear relationship between x and y. Let's also calculate f(0.9) and f(1.1)
x
x^2+4x-5/x-1
f(x)
0.9
( 0.9)^2+4( 0.9)-5/0.9-1
5.9
1.1
( 1.1)^2+4( 1.1)-5/1.1-1
6.1
Finally, we will add these points to the graph from Part A.
Notice that there is no asymptote at x=1. An asymptote is a straight line which a function approaches but never intersects. This is not the case with our function, as the function approaches a point and not a line.
c
To simplify the formula we have to rewrite the numerator by factoring the expression. If the expression can be factored, we should find two terms, a and b, whose sum equals the coefficient to x and whose product equals the constant.
cccll
x^2 & + & 4x &+& -5 [0.3em]
x^2 & + & (a+b)x & +& abLet's factor the product, - 5, in as many ways we can and add the factors. When the sum equals 4, we have identified the correct numbers a and b.
c|c|c|cl
product & a(b) & a+b & sum & [0.2em]
[-1em]
- 5 & - 5(1) & - 5+1& - 4 & * [0.1em]
- 5& - 1(5) & - 1+5& 4 & ✓
By substituting a= - 1 and b= 5 into (x+a)(x+b), we will have factored the expression in the numerator.
(x+( - 1))(x+ 5) ⇔ (x-1)(x+5)
Now we can simplify the formula.
