Problem 25
Question
Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x+4)}$$
Step-by-Step Solution
Verified Answer
The function \(h(x)=\frac{x}{x(x+4)}\) has a vertical asymptote at \(x = -4\) and no holes.
1Step 1: Simplifying the function
First, simplify the function by cancelling out any common factors in the numerator and the denominator. The function \(h(x)=\frac{x}{x(x+4)}\) simplifies to \(h(x)=\frac{1}{x+4}\) because the \(x\)s cancel out.
2Step 2: Solving for the vertical asymptote
A function will have vertical asymptotes where the denominator is equal to zero, provided the factors causing this don’t cancel out. Solve the equation \(x+4 = 0\) for \(x\). The solution \(x = -4\) is a vertical asymptote.
3Step 3: Finding the hole
A function will have a hole where the denominator is equal to zero, but the factors causing this do cancel out. As there are no factors that resulted in division by zero left, there are no holes in the graph of this function.
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