Problem 3
Question
Find the domain of each rational function. $$g(x)=\frac{3 x^{2}}{(x-5)(x+4)}$$
Step-by-Step Solution
Verified Answer
The domain of the function is all real numbers except \(x = 5\) and \(x = -4\).
1Step 1: Express the denominator of the function
We first look at the denominator of \(g(x)\), which is \((x-5)(x+4)\). This product equals zero if either of these terms equals zero.
2Step 2: Find the x values that make the denominator zero
We set each term equal to zero and solve for x. Setting \(x - 5\) equal to zero and solving for x gives \(x = 5\). Setting \(x + 4\) equal to zero and solving for x gives \(x = -4\). These are the values that make the denominator zero.
3Step 3: Exclude these values from the domain
Since the denominator cannot be zero, we exclude these values from the domain. The domain of \(g(x)\) is all real numbers excluding \(x = 5\) and \(x = -4\).
Other exercises in this chapter
Problem 2
Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \left(x^{2}+3 x-10\right) \div(x-2) $$
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Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (x-7
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In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$
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Determine which functions are polynomial functions. For those that are, identify the degree. $$ g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x $$
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