College Algebra  Exercises  Section 4.1

As you study these exercises, move your cursor over the light bulbs and the highlighted words.
Study with an active cursor

The definition that pops up when you move the cursor over a highlighted word is what should pop into your mind when you read, hear, or speak that word. The material that pops up when you move the cursor over a light bulb is a suggested strategy for solving the problem. You should always formulate a similar strategy when you attempt to solve a problem. When you move the cursor over an explanation of a step in a solution, the property that pops up is the mathematical justification for taking that action. You must always be able to provide such a justification for every step you take in mathematics.

5) Consider the function f whose rule is given by . Find its domain, vertical asymptote and horizontal asymptote .
Solution: The domain of f is all real numbers for which the denominator is not zero.
The zero of the denominator is clearly 0.
The domain of f is therefore the set of all non-zero real numbers.
This could be written as

The vertical asymptotes of a rational function are found by finding zeros of the denominator which are not zeros of the numerator.
In this function clearly 0 is a zero of the denominator and just as clearly it is not a zero of the numerator.
Therefore the vertical line x = 0 is a vertical asymptote of the function f.

To find a horizontal asymptote consider
Therefore the horizontal line y = 0 is a horizontal asymptote for the function f.

8) Consider the function f whose rule is given by . Find its domain, vertical asymptote and horizontal asymptote .
Solution: The domain of f is all real numbers for which the denominator is not zero.
The denominator D has the rule D(x) = 1 + 2x
The zero of the denominator D is clearly . 		The numerator N has the rule D(x) = 1 - 5x
The zero of the numerator N is clearly .

The domain of f is therefore the set of all real numbers not equal to .
This could be written as

The vertical asymptotes of a rational function are found by finding zeros of the denominator which are not zeros of the numerator.
The above computation shows that is a zero of the denominator and shows that it is not a zero of the numerator.
Therefore the vertical line x = is a vertical asymptote of the function f.

To find a horizontal asymptote consider
Therefore the horizontal line y = is a horizontal asymptote for the function f.
The computer generated graph at the right shows the asymptotes and the domain of this function.

11) Consider the function f whose rule is given by . Find its domain, vertical asymptote and horizontal asymptote .
Solution: The domain of f is all real numbers for which the denominator is not zero.
The denominator D has the rule D(x) = x2 + x + 9
The zeros of the denominator D may be found with the quadratic formula . But since the discriminant b2 - 4ac = 1 - 36 < 0 the zeros are complex numbers. Therefore the function f is defined for all real numbers and does not have any vertical asymptotes.

To find a horizontal asymptote consider

Therefore the horizontal line y = 3 is a horizontal asymptote for the function f.
The computer generated graph below shows the asymptotes and the domain of this function.

Notice that the graph of f seems to cross its horizontal asymptote around x = -8. But it seems not to cross the horizontal asymptote anywhere else.
The graph does indeed cross the horizontal asymptote y = 3, then approaches that line from above as x approaches negative infinity.
On the other side, the graph does not cross the horizontal asymptote y = 3, but does approach that line from below as x approaches positive infinity.

17) Consider the function f whose rule is given by . Find its zeros .
Solution:
The denominator D has the rule D(x) = x + 1
The zero of the denominator D is clearly -1.
The domain of f is all real numbers other than -1.
We can write this as
The numerator N has the rule N(x) = x2 - 1 = (x - 1)(x + 1)
The zeros of the numerator N are clearly -1 and 1.
-1 is not in the domain of f, and is therefore not a zero of f.
The function f has a single zero, namely 1.

The computer generated graph at the right shows the domain of f and shows that

18) Consider the function f whose rule is given by . Find its zeros .
Solution:
The denominator D has the rule D(x) = x - 3
The zero of the denominator D is clearly 3.
The domain of f is all real numbers other than 3.
We can write this as
The numerator N has the rule
The zeros of the numerator N are clearly - and .
Both of the zeros of the numerator are in the domain of the function f.
Therefore the function f has two zeros - and .

The computer generated graph of f at the right shows the zeros, the domain, and the vertical asymptote at x = 3.

22) Consider the function f whose rule is given by . Find its zeros .
Solution:
In general it is easier to work with the fractional form than the "mixed number" form
The denominator D has the rule D(x) = x + 5
The zero of the denominator D is clearly - 5.
The domain of f is all real numbers other than - 5.
We can write this as
The numerator N has the ruleN(x) = 4x + 18
The zero of the numerator N is clearly .
The zero of the numerator is in the domain of the function f.
Therefore the function f has one zero .

The computer generated graph of f shows the zero, the domain,the vertical asymptote at x = -5, the horizontal asymptote at y = 4 and the asymptotic behavior of the graph of f.