Problem 90 - Algebra and Trigonometry

Question

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x-5}{10 x-2}+\frac{x^{2}-10 x+25}{25 x^{2}-1} $$

Step-by-Step Solution

Answer

The simplified equation for the function is \(f(x) = \frac{35x^{3} -20x^{2} +49x-55 }{250x^{3}-2x-2}\). To graph the function, take note of the x values that cause the denominator to be zero (as these would cause the function to be undefined) and the values of x that cause the numerator to be zero (as these are the x-intercepts of the function).

Step 1: Combine fractions with the same denominator

Here we begin simplifying by combining the two fractions into one. Since they have different denominators, we need to find a common denominator, which is the product of the two original denominators, \(10x - 2\) and \(25x^{2} - 1\). Thus, we multiply the numerator and denominator of each fraction by the denominator of the other, so that both fractions have the same denominator. After this operation, the equation for \(f(x)\) is: \[f(x)=\frac{(x-5)(25x^{2}-1)+(10x-2)(x^{2}-10x+25)}{(10x-2)(25x^{2}-1)}\]

Step 2: Expansion of the Numerator

Now we need to expand the terms in the numerator of the fraction by using distribution:\[f(x)=\frac{25x^{3}-x-5+10x^{3}-20x^{2}+50x-50}{10x(25x^{2}-1)-2(25x^{2}-1)}\]

Step 3: Simplify the function

After combining like terms and doing further simplification of the expression above, we have the function: \[ f(x) = \frac{35x^{3} -20x^{2} +49x-55 }{250x^{3}-2x-2}\]

Step 4: Graph the function

Next step is to graph the function. We cannot provide here the picture of the graph represented by the function, but the it can be done using graphical calculator. Most importantly, the graph of \(f(x)\) will be undefined at x values that make the denominator zero. To find this value, you have to set the denominator equal to zero and solve for \(x\). So, You have to note asymptotes and x-intercepts as well where we set numerator equal to zero and solve for \(x\). It is also beneficial to look for the y-intercept by setting \(x = 0\).

Key Concepts

Common Denominator

To combine fractions, especially in algebraic expressions such as rational functions, finding a common denominator is key. This is akin to making the denominators of all fractions involved identical, allowing you to easily add or subtract them, just like you would numbers. In the given problem, we are dealing with two separate fractions.
A common denominator here is achieved by multiplying the denominators together:

• The first denominator is \(10x - 2\).
• The second denominator is \(25x^2 - 1\).

This means our common denominator is the product \((10x - 2)(25x^2 - 1)\).

• So, modify each fraction by multiplying its numerator and denominator by the opposite denominator.
• This process might seem cumbersome but is necessary for the accurate combination of the fractions into one single expression.

Function Graphing

Graphing functions involves plotting a curve that represents the equation on a graph. For rational functions like the one we simplified, graphing can tell us a lot about the behavior and characteristics of the function.

• Firstly, remember that the graph of a rational function can be undefined where the denominator is zero. This creates vertical asymptotes.
• Using a graphing tool or calculator is advisable for complex equations to visualize these properties accurately.
• Before graphing, determine values where \(f(x)\) is undefined to know what kind of values \(x\) cannot take.

Understanding how to sketch the graph allows us to see intersections with axes, and also the trend of the function as \(x\) approaches infinity.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about breaking down the complexity into a more manageable form. It is a critical step, especially when dealing with rational functions.
In the given exercise, combining the terms inside the numerator after finding a common denominator is imperative:

• This involves multiplying through and gathering like terms.
• Reduction happens by adding or subtracting coefficients of same power terms.

For instance, terms with the same powers of \(x\) should be combined. These manipulation skills are necessary for keeping the function in its simplest form so that calculations become more straightforward.

Asymptotes and Intercepts

Asymptotes and intercepts are essential features of rational functions worth noting.

• Asymptotes:
  • Vertical asymptotes occur where the denominator of the function is zero. No real value of \(f(x)\) exists here, which divides the function into distinct parts.
  • Horizontal or slant asymptotes come from the degree of numerator and denominator, affecting the graph as \(x\) heads to infinity.
• Intercepts:
  • X-intercepts are found by setting the numerator to zero and solving for \(x\).
  • Find the y-intercept by plugging \(x = 0\) into your function, if defined at that point.

Asymptotes guide you on where the graph shoots off and never returns to cross the axis. Intercepts give tangible points where your graph touches or crosses the axes.