If `f (x) =(x+3)/(4x-5) and t =(3+ 5x)/( 4x -1), ` then f (t) is

Options

A. `-x`
B. `17x`
C. `x`
D. `-17x`

Answer

To find \( f(t) \) where \( f(x) = \frac{x + 3}{4x - 5} \) and \( t = \frac{3 + 5x}{4x - 1} \), we will follow these steps:

### Step 1: Substitute \( t \) into \( f(x) \)

We need to replace \( x \) in the function \( f(x) \) with \( t \):
\[ f(t) = f\left(\frac{3 + 5x}{4x - 1}\right) \]

### Step 2: Write the expression for \( f(t) \)

Using the definition of \( f(x) \):
\[ f(t) = \frac{t + 3}{4t - 5} \]

### Step 3: Substitute \( t \) into the expression

Now, we substitute \( t = \frac{3 + 5x}{4x - 1} \) into the expression for \( f(t) \):
\[ f(t) = \frac{\left(\frac{3 + 5x}{4x - 1}\right) + 3}{4\left(\frac{3 + 5x}{4x - 1}\right) - 5} \]

### Step 4: Simplify the numerator

The numerator becomes:
\[ \frac{3 + 5x + 3(4x - 1)}{4x - 1} = \frac{3 + 5x + 12x - 3}{4x - 1} = \frac{17x}{4x - 1} \]

### Step 5: Simplify the denominator

The denominator becomes:
\[ 4\left(\frac{3 + 5x}{4x - 1}\right) - 5 = \frac{12 + 20x}{4x - 1} - 5 = \frac{12 + 20x - 5(4x - 1)}{4x - 1} = \frac{12 + 20x - 20x + 5}{4x - 1} = \frac{17}{4x - 1} \]

### Step 6: Combine the numerator and denominator

Now we can combine the simplified numerator and denominator:
\[ f(t) = \frac{\frac{17x}{4x - 1}}{\frac{17}{4x - 1}} = x \]

### Final Result
Thus, we find that:
\[ f(t) = x \]