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Chapter 0: Problem 66

\(h(x)=\frac{10}{x^{2}-2 x}\)

Short Answer

Expert verified
The simplified form of the function is \(h(x)=\frac{10}{x(x-2)}\) which is a rational function and is undefined for x=0 and x=2.

Step by step solution

01

Factorize the Denominator

Rearrange the denominator \(x^{2}-2x\) by factoring out the common factor x from both terms. This will give you: \(x(x - 2)\).
02

Rewrite the Function

Rewrite the original function using the factorized form of the denominator. Now, the function will look like this: \(h(x)=\frac{10}{x(x-2)}\).
03

Analyze the Function

The simplified function represents a rational function. It is important to note that the function is undefined for x=0 and x=2 as these values would make the denominator zero.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied together, give back the original polynomial.

For the given expression, you have the denominator as \(x^2 - 2x\).

To factor this, look for the greatest common factor (GCF), which is a term that can be pulled out from every part of the expression.
  • In \(x^2\), the term \(x\) can be extracted.
  • Similarly, in \(2x\), you can factor out \(x\).

This results in the following factorization: \(x(x - 2)\).

Factoring helps to simplify complex expressions and is particularly useful when dealing with rational functions, where identifying removable singularities鈥攙alues that create zero denominators鈥攃omes into play.
Domain of a Function
The domain of a function is the set of all possible input values (in this case, \(x\)) that will yield valid outputs.

For rational functions, like \(h(x) = \frac{10}{x(x-2)}\), finding the domain involves checking where the denominator is not equal to zero, since division by zero is undefined.
  • Set each factor of the denominator equal to zero: \(x = 0\) and \(x - 2 = 0\).
  • This gives the solutions \(x = 0\) and \(x = 2\), which are not part of the domain.

Excluding \(x = 0\) and \(x = 2\), the domain of \(h(x)\) is all real numbers except these points.

In interval notation, this domain is represented as \((-\infty, 0) \cup (0, 2) \cup (2, \infty)\).
Identifying the domain is crucial, as it tells us the permissible values \(x\) can take without making the function undefined.
Undefined Expressions
In mathematics, expressions become undefined when they result in mathematical impossibilities or indeterminacies, such as division by zero.

For our function, \(h(x) = \frac{10}{x(x-2)}\), both \(x = 0\) and \(x = 2\) make the denominator zero.
  • When \(x = 0\), the expression becomes \(\frac{10}{0}\), which is undefined.
  • Similarly, when \(x = 2\), the outcome is \(\frac{10}{0}\), again resulting in an undefined expression.

These points where the expression becomes undefined are called singularities of the function.

Recognizing when an expression can become undefined is vital, especially in calculus, as it affects function limits, graphs, and the integrability of the function.
Understanding undefined expressions can prevent errors in solving equations and graphing rational functions.

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{1}{x^{2}} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=3 x+5 $$

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ g(s)=4 s^{2 / 3} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{5}-2 $$
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