/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 90 The equation for \(f\) is given ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 90

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

Short Answer

Expert verified
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 卤鈭� based on its degree and leading coefficient.

Step by step solution

01

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: \(\frac{x-5}{10x-2} * \frac{25x^{2}-1}{x^{2}-10x+25}\)
02

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: \(f(x) = \frac{5x+1}{2(x-5)}\)
03

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 卤鈭� based on its degree and leading coefficient

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Most popular questions from this chapter

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{x+2}{x-3} \leq 2 $$

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I like to think of a parabola's vertex as the point where it intersects its axis of symmetry.

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