If g(x)=x^(2) +x−1 and (gof)(x)=4x^(2) −10x+5, then f(5/4 ​ ) is equal to:

Options

A. a. 3/2
B. b. -1/2
C. c. -3/2
D. d. none of these

Answer

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

### Step-by-step Solution:

1. **Understand the Composition of Functions:**
The expression \( (g \circ f)(x) \) means \( g(f(x)) \). This means we will substitute \( f(x) \) into the function \( g(x) \).

2. **Set Up the Equation:**
We know that:
\[ g(f(x)) = 4x^2 - 10x + 5 \]
and we also know:
\[ g(x) = x^2 + x - 1 \]
Therefore, we can write:
\[ g(f(x)) = f(x)^2 + f(x) - 1 \]

3. **Equate the Two Expressions:**
Set the two expressions for \( g(f(x)) \) equal to each other:
\[ f(x)^2 + f(x) - 1 = 4x^2 - 10x + 5 \]

4. **Rearrange the Equation:**
Rearranging gives us:
\[ f(x)^2 + f(x) - (4x^2 - 10x + 6) = 0 \]
This simplifies to:
\[ f(x)^2 + f(x) - 4x^2 + 10x - 6 = 0 \]

5. **Use the Quadratic Formula:**
The above is a quadratic equation in terms of \( f(x) \):
\[ a = 1, \quad b = 1, \quad c = - (4x^2 - 10x + 6) \]
The quadratic formula is:
\[ f(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting \( a, b, c \):
\[ f(x) = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-(4x^2 - 10x + 6))}}{2 \cdot 1} \]
This simplifies to:
\[ f(x) = \frac{-1 \pm \sqrt{1 + 16x^2 - 40x + 24}}{2} \]
\[ = \frac{-1 \pm \sqrt{16x^2 - 40x + 25}}{2} \]

6. **Simplify the Square Root:**
The expression inside the square root can be factored:
\[ 16x^2 - 40x + 25 = (4x - 5)^2 \]
Thus, we have:
\[ f(x) = \frac{-1 \pm (4x - 5)}{2} \]

7. **Find the Two Possible Solutions:**
This gives us two potential solutions:
\[ f(x) = \frac{4x - 5 - 1}{2} = \frac{4x - 6}{2} = 2x - 3 \]
or
\[ f(x) = \frac{-4x + 5 - 1}{2} = \frac{-4x + 4}{2} = -2x + 2 \]

8. **Evaluate \( f\left(\frac{5}{4}\right) \):**
We will evaluate both functions at \( x = \frac{5}{4} \):
- For \( f(x) = 2x - 3 \):
\[ f\left(\frac{5}{4}\right) = 2 \cdot \frac{5}{4} - 3 = \frac{10}{4} - 3 = \frac{10}{4} - \frac{12}{4} = -\frac{2}{4} = -\frac{1}{2} \]
- For \( f(x) = -2x + 2 \):
\[ f\left(\frac{5}{4}\right) = -2 \cdot \frac{5}{4} + 2 = -\frac{10}{4} + \frac{8}{4} = -\frac{2}{4} = -\frac{1}{2} \]

9. **Conclusion:**
In both cases, we find:
\[ f\left(\frac{5}{4}\right) = -\frac{1}{2} \]

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