In applications, variables may occur in the numerator or denominator of a fraction, or both. For example, the time it takes for the Moon to eclipse the Sun is given by the formula
\begin{equation*}
T = \dfrac{rD-Rd}{vR}
\end{equation*}
where the variables give the diameter of the Sun and the distances of the Moon and Sun from Earth. The total resistance of two appliances with resistances \(a\) and \(b\) connected in parallel is given by
These two formulas involve simple poynomials in more than one variable. Letβs start our study of fractions with expressions that include only one variable.
If we try to evaluate the fraction \(\dfrac{a^2+1}{a-2}\) for \(a=2\text{,}\) we get \(\dfrac{2^2+1}{2-2}\) or \(\dfrac{5}{0}\text{,}\) which is undefined. When working with fractions, we must exclude any values of the variable that make the denominators equal to zero.
Youβll recall from your study of arithmetic that we can reduce a fraction if we can divide both numerator and denominator by a common factor. In algebra, it is helpful to think of factoring out the common factor first. For example,
where we have divided both numerator and denominator by 9. The new fraction has the same value as the old one, namely 0.75, but it is simpler (the numbers are smaller.) Reducing is an application of the Fundamental Principle of Fractions.
We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.
The common factor for numerator and denominator is \(2x^2y\text{.}\) We factor \(2x^2y\) from the numerator and denominator, then divide by the common factor.
When we cancel common factors, we are dividing. Because division is the inverse or opposite operation for multiplication, we can cancel common factors, but we cannot cancel common terms.
If the numerator or denominator of the fraction contains more than one term, it is especially important to factor before reducing, so that numerator and denominator are written as products of factors, instead of as sums of terms.
Then we divide numerator and denominator by the common factors 3 and \((x+4)\text{.}\) We must cancel the entire expression \((x+4)\) from numerator and denominator (we cannot cancel the \(x\)βs or the 4βs separately!).
Keep in mind that the reduced form is equivalent to the original form of the fraction. If we evaluate the original form and the reduced form at the same value of the variable, the results are equal.
The same is true for binomials and other algebraic expressions. The opposite of an expression can be found by multiplying it by \(-1\text{.}\) Thus, the opposite of \(a-b\) is
\begin{equation*}
-(a-b) = -a + b= b-a
\end{equation*}
Francine is planning a 60-mile training flight through the desert on her cycle-plane, a pedal-driven aircraft. If there is no wind, she can pedal at an average speed of 15 miles per hour, so she can complete the flight in 4 hours.
If there is a headwind of \(x\) miles per hour, it will take Francine longer to fly 60 miles. Express the time it will take to complete the training flight as a function of \(x\text{.}\)
If there is a headwind of \(x\) miles per hour, Francineβs ground speed will be \(15 - x\) miles per hour. Using the fact that \(\text{time} = \dfrac{\text{distance}}{\text{rate}}\text{,}\) we find that the time needed for the flight will be
\begin{equation*}
t = f(x) = \frac{60}{15 - x}
\end{equation*}
Francineβs effective speed is only 10 miles per hour, and it will take her 6 hours to fly the 60 miles. The table shows that as the speed of the headwind increases, the time required for the flight increases also.
to verify the graph. In particular, the point \((0,4)\) lies on the graph. This point tells us that if there is no wind, Francine can fly 60 miles in 4 hours, as we calculated earlier.
The graph is increasing, as indicated by the table of values. In fact, as the speed of the wind gets close to 15 miles per hour, Francineβs flying time becomes extremely large. In theory, if the wind speed were exactly 15 miles per hour, Francine would never complete her flight. On the graph, the time becomes infinite at \(x = 15\text{.}\)
What about negative values for \(x\text{?}\) If we interpret a negative headwind as a tailwind, Francineβs flying time should decrease for negative \(x\)-values. For example, if \(x = -5\text{,}\) there is a tailwind of 5 miles per hour, so Francineβs effective speed is 20 miles per hour, and she can complete the flight in 3 hours. As the tailwind gets stronger (that is, as we move farther to the left in the \(x\)-direction), Francineβs flying time continues to decrease, and the graph approaches the \(x\)-axis.
The vertical dashed line at \(x=15\) on the graph of \(t=\dfrac{60}{15-x}\) is a vertical asymptote for the graph. We first encountered asymptotes in SubsectionΒ 5.3.3 when we studied the graph of \(y=\dfrac{1}{x}\text{.}\) Locating the vertical asymptotes of a rational function is an important part of determining the shape of the graph.
EarthCare decides to sell T-shirts to raise money. The company makes an initial investment of $100 to pay for the design of the T-shirt and to set up the printing process. After that, the T-shirts cost $5 each for labor and materials.
In Practice 7, the horizontal line \(C=5\) is a horizontal asymptote for the graph of the function. As \(x\) increases, the graph approaches the line \(C=5\) but will never actually meet it. The average price per T-shirt will always be slightly more than $5. Horizontal asymptotes are also important in sketching the graphs of rational functions. \(~\alert{\text{[TK]}}\)
The crew team can row at a steady pace of 10 miles per hour in still water. Every afternoon, their training includes a five-mile row upstream on the river. If the current in the river on a given day is \(v\) miles per hour, then the time required for this exercise, in minutes, is given by
The eider duck, one of the worldβs fastest flying birds, can exceed an airspeed of 65 miles per hour. A flock of eider ducks is migrating south at an average airspeed of 50 miles per hour against a moderate headwind. Their next feeding grounds are 150 miles away.
Express the ducksβ travel time, \(t\text{,}\) as a function of the windspeed, \(v\text{.}\)
The volume of a test tube is given by its height times the area of its cross-section. A test tube that holds 200 cubic centimeters is \(2x-1\) centimeters long.
Delbert prepares a 25% glucose solution of by mixing 2 ml of glucose with 6 ml of water. If he adds \(x\) ml of glucose to the solution, its concentration is given by
A computer store sells approximately 300 of its most popular model per year. The manager would like to minimize her annual inventory cost by ordering the optimal number of computers, \(x\text{,}\) at regular intervals. If she orders \(x\) computers in each shipment, the cost of storage will be \(6x\) dollars, and the cost of reordering will be \(\dfrac{300}{x} (15x + 10)\) dollars. The inventory cost is the sum of the storage cost and the reordering cost.
Use the distributive law to simplify the expression for the reordering cost. Then express the inventory cost, \(C\text{,}\) as a function of \(x\text{.}\)
A train whistle sounds higher when the train is approaching you than when it is moving away from you. This phenomenon is known as the Doppler effect. If the actual pitch of the whistle is 440 hertz (this is the A note below middle C), then the note you hear will have the pitch
where the velocity, \(v\text{,}\) in meters per second is positive as the train approaches and is negative when the train is moving away. (The number 332 that appears in this expression is the speed of sound in meters per second.)
Complete the table of values showing the pitch of the whistle at various train velocities.
What is the velocity of the train if the note you hear has a pitch of 415 hertz (corresponding to the note A-flat)? A pitch of \(553.\overline{3}\) hertz (C-sharp)?