Find the vertical asymptote(s) of the graph of the function.

Ask by Rojas Parsons. in the United States

Mar 25,2025

Question

Find the vertical asymptote(s) of the graph of the function. $$ f(x)=\frac{2-x}{(x-3)(x+2)} $$

UpStudy AI · Accepted Answer

1. **Find the values of $$ x $$ that make the denominator zero.**  
   The denominator of the function is $$(x-3)(x+2)$$. Setting it equal to zero gives:  
   $$
   (x-3)(x+2)=0
   $$
   Therefore,  
   $$
   x-3=0 \quad \text{or} \quad x+2=0.
   $$

2. **Solve each equation.**  
   For $$ x-3=0 $$:  
   $$
   x=3.
   $$  
   For $$ x+2=0 $$:  
   $$
   x=-2.
   $$

3. **Verify that these values do not cancel with factors of the numerator.**  
   The numerator is $$ 2-x $$. Evaluate the numerator at $$ x=3 $$ and $$ x=-2 $$:
   - At $$ x=3 $$:  
     $$
     2-3=-1 \quad (\text{non-zero}).
     $$
   - At $$ x=-2 $$:  
     $$
     2-(-2)=4 \quad (\text{non-zero}).
     $$
   Since the numerator is non-zero at both $$ x=3 $$ and $$ x=-2 $$, these points are not removable discontinuities.

4. **Conclusion:**  
   The vertical asymptotes of the function are at  
   $$
   x=3 \quad \text{and} \quad x=-2.
   $$
The vertical asymptotes are at $$ x=3 $$ and $$ x=-2 $$.

Find the vertical asymptote(s) of the graph of the function.

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