If `g(x)=x^(2)+x-1 and g(f(x))=4x^(2)-10x+5` then find `f((5)/(4))`

Options

A. 1. `3/2`
B. 2.`-(3)/(2)`
C. 3. `1/2`
D. 4. `-(1)/(2)`

Answer

To solve the problem, we need to find the value of \( f\left(\frac{5}{4}\right) \) given that \( g(x) = x^2 + x - 1 \) and \( g(f(x)) = 4x^2 - 10x + 5 \).

### Step-by-Step Solution:

1. **Understanding the Functions**:
We have:
\[ g(x) = x^2 + x - 1 \]
and
\[ g(f(x)) = 4x^2 - 10x + 5. \]

2. **Substituting \( x = \frac{5}{4} \)**:
We need to calculate \( g(f\left(\frac{5}{4}\right)) \):
\[ g(f\left(\frac{5}{4}\right)) = 4\left(\frac{5}{4}\right)^2 - 10\left(\frac{5}{4}\right) + 5. \]

3. **Calculating \( g(f\left(\frac{5}{4}\right)) \)**:
First, calculate \( 4\left(\frac{5}{4}\right)^2 \):
\[ 4\left(\frac{25}{16}\right) = \frac{100}{16} = \frac{25}{4}. \]
Next, calculate \( -10\left(\frac{5}{4}\right) \):
\[ -10\left(\frac{5}{4}\right) = -\frac{50}{4} = -\frac{50}{4}. \]
Now add 5 (which is \( \frac{20}{4} \)):
\[ g(f\left(\frac{5}{4}\right)) = \frac{25}{4} - \frac{50}{4} + \frac{20}{4} = \frac{25 - 50 + 20}{4} = \frac{-5}{4}. \]

4. **Setting up the Equation**:
We have:
\[ g(f\left(\frac{5}{4}\right)) = f\left(\frac{5}{4}\right)^2 + f\left(\frac{5}{4}\right) - 1. \]
Therefore, we can set up the equation:
\[ f\left(\frac{5}{4}\right)^2 + f\left(\frac{5}{4}\right) - 1 = -\frac{5}{4}. \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ f\left(\frac{5}{4}\right)^2 + f\left(\frac{5}{4}\right) + \frac{1}{4} = 0. \]

6. **Recognizing the Form**:
The left-hand side can be recognized as a perfect square:
\[ \left(f\left(\frac{5}{4}\right) + \frac{1}{2}\right)^2 = 0. \]
Hence, we can conclude:
\[ f\left(\frac{5}{4}\right) + \frac{1}{2} = 0. \]

7. **Solving for \( f\left(\frac{5}{4}\right) \)**:
Thus,
\[ f\left(\frac{5}{4}\right) = -\frac{1}{2}. \]

### Final Answer:
\[ f\left(\frac{5}{4}\right) = -\frac{1}{2}. \]