Let f (x) = x/(x − 6). Answer the following:
a) f (4) =   b) f (1/2) =
2
c) f (t − 1) = d) f (t + 6) =
e) f (2/y) = f) f (3/(y + 5)) =

We are given the function \( f(x) = \frac{x}{x - 6} \). Let's evaluate the function for each of the given values.

### a) \( f(4) \)
Substitute \( x = 4 \) into \( f(x) \):

\[
f(4) = \frac{4}{4 - 6} = \frac{4}{-2} = -2
\]

### b) \( f\left(\frac{1}{2}\right) \)
Substitute \( x = \frac{1}{2} \) into \( f(x) \):

\[
f\left(\frac{1}{2}\right) = \frac{\frac{1}{2}}{\frac{1}{2} - 6} = \frac{\frac{1}{2}}{\frac{1}{2} - \frac{12}{2}} = \frac{\frac{1}{2}}{-\frac{11}{2}} = \frac{1}{2} \times \frac{2}{-11} = \frac{1}{-11} = -\frac{1}{11}
\]

### c) \( f(t - 1) \)
Substitute \( x = t - 1 \) into \( f(x) \):

\[
f(t - 1) = \frac{t - 1}{(t - 1) - 6} = \frac{t - 1}{t - 7}
\]

### d) \( f(t + 6) \)
Substitute \( x = t + 6 \) into \( f(x) \):

\[
f(t + 6) = \frac{t + 6}{(t + 6) - 6} = \frac{t + 6}{t}
\]

### e) \( f\left(\frac{2}{y}\right) \)
Substitute \( x = \frac{2}{y} \) into \( f(x) \):

\[
f\left(\frac{2}{y}\right) = \frac{\frac{2}{y}}{\frac{2}{y} - 6} = \frac{\frac{2}{y}}{\frac{2}{y} - \frac{6y}{y}} = \frac{\frac{2}{y}}{\frac{2 - 6y}{y}} = \frac{2}{2 - 6y}
\]

### f) \( f\left(\frac{3}{y + 5}\right) \)
Substitute \( x = \frac{3}{y + 5} \) into \( f(x) \):

\[
f\left(\frac{3}{y + 5}\right) = \frac{\frac{3}{y + 5}}{\frac{3}{y + 5} - 6} = \frac{\frac{3}{y + 5}}{\frac{3}{y + 5} - \frac{6(y + 5)}{y + 5}} = \frac{\frac{3}{y + 5}}{\frac{3 - 6(y + 5)}{y + 5}} = \frac{3}{3 - 6(y + 5)}
\]

Simplify the denominator:

\[
3 - 6(y + 5) = 3 - 6y - 30 = -6y - 27
\]

Thus, 

\[
f\left(\frac{3}{y + 5}\right) = \frac{3}{-6y - 27}
\]

---

Would you like more details on any of these calculations?

Here are five related questions:
1. What is the domain of \( f(x) = \frac{x}{x - 6} \)?
2. How do you find vertical asymptotes for rational functions like \( f(x) \)?
3. Can you simplify \( f(t + 6) \) further?
4. What happens if \( y = -5 \) in \( f\left(\frac{3}{y + 5}\right) \)?
5. How do you evaluate the limits of \( f(x) \) as \( x \to 6 \)?

**Tip:** When simplifying fractions, always try to factor the denominator to see if anything can be canceled out.

Let f(x) = x/(x − 6). Answer the following: a) f(4) =   b) f(1/2) =   c) f(t − 1) =   d) f(t + 6) =   e) f(2/y) =   f) f(3/(y + 5)) =

Solve the function f(x) = x / (x - 6) for various inputs like f(4), f(1/2), f(t-1), and more. This problem demonstrates key algebraic concepts such as evaluating rational functions and simplifying expressions. Suitable for Grades 10-12.

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