Problem 90 - College Algebra

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. $$\frac{x-5}{10 x-2} \div \frac{x^{2}-10 x+25}{25 x^{2}-1}$$

Step-by-Step Solution

Answer

The simplified expression for \(f\) is \(f(x) = \frac{5x+1}{2(x-5)}\). The graph of the function has vertical asymptote at \(x = 5\) and horizontal asymptote at \(y = 0.5\), and it behaves correctly towards ±∞ based on its degree and leading coefficient.

Step 1: Simplify the function

Firstly, divide \(\frac{x-5}{10x-2}\) by \(\frac{x^{2}-10x+25}{25x^{2}-1}\). It is helpful to change division to multiplication, which also involves inverting the second fraction: \(\frac{x-5}{10x-2} * \frac{25x^{2}-1}{x^{2}-10x+25}\)

Step 2: Further simplification

The numerator and the denominator can be factored as follows: \(\frac{x-5}{2(5x-1)} * \frac{(5x-1)(5x+1)}{(x-5)^2}\). Notice that there are common terms in the numerator and the denominator, they can be cancelled out. Therefore, the simplified function is: \(f(x) = \frac{5x+1}{2(x-5)}\)

Step 3: Graphing the function

To graph this function, you need to identify key characteristics of the function. The graph will have a vertical asymptote at \(x = 5\) and a horizontal asymptote at \(y = 0.5\). To graph these elements, plot a number of points on either side of the asymptote, plot the asymptotes, then draw the graph approaching the asymptotes. Ensure the graph behaves correctly towards ±∞ based on its degree and leading coefficient