If `f(x)=2x^(2)+6x-1`, then the value of `(f((3)/(4))+1)/(f((3)/(4))-1)` is

Options

A. `11//13`
B. `35//3`
C. `45//29`
D. None of these

Answer

To solve the problem, we need to evaluate the expression \((f(\frac{3}{4}) + 1) / (f(\frac{3}{4}) - 1)\) given the function \(f(x) = 2x^2 + 6x - 1\).

### Step 1: Calculate \(f(\frac{3}{4})\)

We start by substituting \(\frac{3}{4}\) into the function \(f(x)\):

\[ f\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 + 6\left(\frac{3}{4}\right) - 1 \]

### Step 2: Simplify \(\left(\frac{3}{4}\right)^2\)

Calculating \(\left(\frac{3}{4}\right)^2\):

\[ \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Step 3: Substitute back into the function

Now substitute \(\frac{9}{16}\) back into the function:

\[ f\left(\frac{3}{4}\right) = 2 \cdot \frac{9}{16} + 6 \cdot \frac{3}{4} - 1 \]

### Step 4: Calculate \(2 \cdot \frac{9}{16}\)

Calculating \(2 \cdot \frac{9}{16}\):

\[ 2 \cdot \frac{9}{16} = \frac{18}{16} = \frac{9}{8} \]

### Step 5: Calculate \(6 \cdot \frac{3}{4}\)

Calculating \(6 \cdot \frac{3}{4}\):

\[ 6 \cdot \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \]

### Step 6: Combine all parts

Now combine all parts:

\[ f\left(\frac{3}{4}\right) = \frac{9}{8} + \frac{9}{2} - 1 \]

### Step 7: Convert \(\frac{9}{2}\) and \(1\) to have a common denominator

Convert \(\frac{9}{2}\) and \(1\) to have a common denominator of 8:

\[ \frac{9}{2} = \frac{36}{8}, \quad 1 = \frac{8}{8} \]

### Step 8: Combine fractions

Now combine:

\[ f\left(\frac{3}{4}\right) = \frac{9}{8} + \frac{36}{8} - \frac{8}{8} = \frac{9 + 36 - 8}{8} = \frac{37}{8} \]

### Step 9: Substitute \(f(\frac{3}{4})\) into the expression

Now substitute \(f(\frac{3}{4})\) into the expression:

\[ \frac{f\left(\frac{3}{4}\right) + 1}{f\left(\frac{3}{4}\right) - 1} = \frac{\frac{37}{8} + 1}{\frac{37}{8} - 1} \]

### Step 10: Simplify the numerator and denominator

Calculate the numerator:

\[ \frac{37}{8} + 1 = \frac{37}{8} + \frac{8}{8} = \frac{45}{8} \]

Calculate the denominator:

\[ \frac{37}{8} - 1 = \frac{37}{8} - \frac{8}{8} = \frac{29}{8} \]

### Step 11: Final calculation

Now we can simplify the expression:

\[ \frac{\frac{45}{8}}{\frac{29}{8}} = \frac{45}{29} \]

Thus, the final answer is:

\[ \frac{45}{29} \]