Rational Function Detective Worksheet
Function Detective
Solving Rational Equations with Tables
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Date:
Investigation Instructions
Analyze the rational function and its table of values for each problem. Use the provided data to determine which statement is true about the solution to the given equation.
1
Some values of the function \( f(x) = \frac{2}{x+3} \) are shown in the table below.
| \( x \) | \( f(x) \) |
|---|
| -5 | -1 |
| -4 | -2 |
| -3.5 | -4 |
| -2 | 2 |
| -1 | 1 |
| 0 | 0.667 |
Which of the following is true about the solution to the equation \( \frac{2}{x+3} = 1 \)?
A The solution is -1.
B The solution is 0.667.
C The solution is between -3 and -2.
D There is no solution to the equation.
2
Some values of the function \( g(x) = \frac{1}{4-x} \) are shown in the table below.
| \( x \) | \( g(x) \) |
|---|
| 1 | 0.333 |
| 2 | 0.5 |
| 3 | 1 |
| 3.5 | 2 |
| 4.5 | -2 |
| 5 | -1 |
Which of the following is true about the solution to the equation \( \frac{1}{4-x} = -1 \)?
A The solution is 5.
B The solution is -1.
C The solution is between 3 and 4.
D The solution is 3.5.
3
Some values of the function \( h(x) = \frac{3}{2x-1} \) are shown in the table below.
| \( x \) | \( h(x) \) |
|---|
| -1 | -1 |
| 0 | -3 |
| 0.5 | undef. |
| 1 | 3 |
| 2 | 1 |
| 3 | 0.6 |
Which of the following is true about the solution to the equation \( \frac{3}{2x-1} = 3 \)?
A The solution is 1.
B The solution is -1.
C The solution is 3.
D The solution is 0.5.
4
Some values of the function \( j(x) = \frac{-4}{x+1} \) are shown in the table below.
| \( x \) | \( j(x) \) |
|---|
| -5 | 1 |
| -3 | 2 |
| -2 | 4 |
| 0 | -4 |
| 1 | -2 |
| 3 | -1 |
Which of the following is true about the solution to the equation \( \frac{-4}{x+1} = 2 \)?
A The solution is -4.
B The solution is -3.
C The solution is between 0 and 1.
D The solution is 1.
5
Some values of the function \( k(x) = \frac{1}{3x+2} \) are shown in the table below.
| \( x \) | \( k(x) \) |
|---|
| -2 | -0.25 |
| -1 | -1 |
| 0 | 0.5 |
| 0.33 | 0.33 |
| 1 | 0.2 |
| 2 | 0.125 |
Which of the following is true about the solution to the equation \( \frac{1}{3x+2} = -1 \)?
A The solution is -1.
B The solution is 0.33.
C The solution is between -2 and -1.
D There is no solution to the equation.
Rational Function Detective Answer Key
Function Detective
Teacher Answer Key
Answer Key
Key Concepts
- Students should identify the target output value in the \( f(x) \) column.
- The corresponding \( x \) value in that row is the solution to the equation.
- Vertical asymptotes are marked as "undef." in tables.
1
Some values of the function \( f(x) = \frac{2}{x+3} \) are shown in the table below.
| \( x \) | \( f(x) \) |
|---|
| -5 | -1 |
| -4 | -2 |
| -3.5 | -4 |
| -2 | 2 |
| -1 | 1 |
| 0 | 0.667 |
Which of the following is true about the solution to the equation \( \frac{2}{x+3} = 1 \)?
A The solution is -1.
B The solution is 0.667.
C The solution is between -3 and -2.
D There is no solution to the equation.
2
Some values of the function \( g(x) = \frac{1}{4-x} \) are shown in the table below.
| \( x \) | \( g(x) \) |
|---|
| 1 | 0.333 |
| 2 | 0.5 |
| 3 | 1 |
| 3.5 | 2 |
| 4.5 | -2 |
| 5 | -1 |
Which of the following is true about the solution to the equation \( \frac{1}{4-x} = -1 \)?
A The solution is 5.
B The solution is -1.
C The solution is between 3 and 4.
D The solution is 3.5.
3
Some values of the function \( h(x) = \frac{3}{2x-1} \) are shown in the table below.
| \( x \) | \( h(x) \) |
|---|
| -1 | -1 |
| 0 | -3 |
| 0.5 | undef. |
| 1 |
