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Chapter 0: Problem 20

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x}\), and \(h(x)=\frac{x+1}{3 x-1}\). Express the following as rational functions. $$ f(x+2)+g(x+2) $$

Short Answer

Expert verified
f(x+2)+g(x+2) = \frac{12x + 14}{x(x+7)}.

Step by step solution

01

Substitute in the functions

Substitute \(x+2\) into the functions \(f(x)\) and \(g(x)\). This will yield \(f(x+2)=\frac{x+2}{(x+2)-2}=\frac{x+2}{x}\) and \(g(x+2)=\frac{5-(x+2)}{5+(x+2)}=\frac{3-x}{x+7}\).
02

Add the two rational functions

Add the two substituted functions from Step 1: \[f(x+2) + g(x+2) = \frac{x+2}{x} + \frac{3-x}{x+7}.\]
03

Find common denominator

Multiply the numerators and denominators of both fractions by a term that will produce a common denominator. Here, the common denominator is \(x(x+7)\).\[f(x+2) + g(x+2) = \frac{(x+2)(x+7) + (3-x)x}{x(x+7)}.\]
04

Simplify the numerator

Expand the numerator and combine like terms to simplify.\[(x+2)(x+7) = x^2 + 9x + 14\]\[(3-x)x = 3x - x^2\]Combine these, we get:\[f(x+2) + g(x+2) = \frac{x^2 + 9x + 14 + 3x - x^2}{x(x+7)}= \frac{12x + 14}{x(x+7)}.\]
05

Factor and simplify if possible

Factor out common terms if possible. Here, \(14=2*7\) so the simplified version is \[f(x+2) + g(x+2) = \frac{2(6x+7)}{x(x+7)}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Substitution
Function substitution is a method used to evaluate functions at specific points or to transform one function into another. In our exercise, we were given functions involving variables. We then substituted the variable with a different expression, specifically \( x+2 \), into each function. The steps let us rewrite \ f(x) \ and \ g(x) \ as follows:

\( f(x+2) = \frac{x+2}{x} \) and \( g(x+2) = \frac{3-x}{x+7} \).

This way, we can manipulate and explore the functions in a new context, simplifying the analysis.
Common Denominator
The concept of a common denominator is crucial when adding or comparing fractions. In this exercise, we needed to add two rational functions:

\( \frac{x+2}{x} + \frac{3-x}{x+7} \).

To add these fractions, we first found a common denominator that both terms could share. A common denominator in fraction addition is a multiple of the denominators of individual fractions. We used:

\( x(x+7) \)

Multiplying the numerators and denominators of each fraction by terms that gave this common denominator:

\( \frac{(x+2)(x+7) + (3-x)x}{x(x+7)} \)

Now, our fractions are ready to be combined.
Simplification
Simplification entails reducing an expression to its simplest form. After finding a common denominator, we expanded the numerators:

\( (x+2)(x+7) = x^2 + 9x + 14 \)
\( (3-x)x = 3x - x^2 \)

Combining these terms resulted in:

\( x^2 + 9x + 14 + 3x - x^2 \)

Then, we simplified this:^+ 9x + 14 + 3x - x^2 \)\(. Combining like terms, we get:

\( 12x + 14 \)
This creates a much simpler expression for easier management and interpretation.
Fraction Addition
Fraction addition with rational functions follows the same principles as adding simple fractions. After determining a common denominator, we combined the numerators:

\( \frac{x+2}{x} + \frac{3-x}{x+7} \rightarrow \frac{(x+2)(x+7) + (3-x)x}{x(x+7)} \).

Combining and simplifying the numerators gave us:

\( \frac{12x + 14}{x(x+7)} \).

Finally, we factored out common terms (if applicable). Here, factoring wasn't necessary, but recognizing that:

\( 14 = 2 \cdot 7 \) is useful if further simplification is required.

Hence, the final, rational function form of the sum is:

\( \frac{2(6x + 7)}{x(x+7)} \).

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Most popular questions from this chapter

Compute the numbers. \(\left(\frac{1}{8}\right)^{-2 / 3}\)

Let \(f(x)=x^{2}+3 x+1\) and let \(g(x)=x^{2}-3 x-1\). Graph the two functions \(f(g(x))\) and \(g(f(x))\) together in the window \([-4,4]\) by \([-10,10]\) and determine if they are the same function.

Evaluate \(f(4)\). \(f(x)=x^{-5 / 2}\)

Compute the numbers. \(9^{1.5}\)

Velocity When a car is moving at \(x\) miles per hour and the driver decides to slam on the brakes, the car will travel \(x+\frac{1}{20} x^{2}\) feet. (The general formula is \(f(x)=a x+b x^{2}\), where the constant \(a\) depends on the driver's reaction time and the constant \(b\) depends on the weight of the car and the type of tires.) If a car travels 175 feet after the driver decides to stop, how fast was the car moving? (Source: Applying Mathematics: A Course in Mathematical Modelling.)
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