Questions: Compare and contrast y=f(x) with y=1/f(x) by completing the chart below. (4 marks) y=f(x) y=1/f(x) ------ The value of the function is 1 The value of the reciprocal is Function values are negative The values of the reciprocal function are negative. The values of the function are greater than 1. Value of function is -1 The value of the reciprocal function is -1. The graph shows a vertical asymptote at this x value. The absolute value of the function approaches infinity. Reciprocal The value of the function is 0. The absolute value of the reciprocal function approaches zero.

Solution Steps

We are given two functions: \( y = f(x) \) and its reciprocal \( y = \frac{1}{f(x)} \). We need to compare and contrast these functions by completing the chart.

1. The value of the function is 1:

   • If \( f(x) = 1 \), then \( \frac{1}{f(x)} = \frac{1}{1} = 1 \).
   • The value of the reciprocal is 1.

2. Function values are negative:

   • If \( f(x) < 0 \), then \( \frac{1}{f(x)} < 0 \).
   • The values of the reciprocal function are negative. (This is already filled in.)

3. The values of the function are greater than 1:

   • If \( f(x) > 1 \), then \( \frac{1}{f(x)} < 1 \).
   • The values of the reciprocal function are less than 1.

4. Value of function is -1:

   • If \( f(x) = -1 \), then \( \frac{1}{f(x)} = \frac{1}{-1} = -1 \).
   • The value of the reciprocal function is \(-1\). (This is already filled in.)

5. The graph shows a vertical asymptote at this \(x\) value:

   • A vertical asymptote occurs when \( f(x) = 0 \) because \( \frac{1}{0} \) is undefined.
   • The graph shows a vertical asymptote at this \(x\) value when \( f(x) = 0 \).

6. The absolute value of the function approaches infinity:

   • As \( |f(x)| \to \infty \), \( \left|\frac{1}{f(x)}\right| \to 0 \).
   • The absolute value of the reciprocal function approaches zero.

7. The value of the function is 0:

   • If \( f(x) = 0 \), then \( \frac{1}{f(x)} \) is undefined.
   • The reciprocal function is undefined.

Final Answer