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

Options

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

Answer

To solve the problem, we start with the given functions:

1. \( g(x) = x^2 + x - 1 \)
2. \( g(f(x)) = 4x^2 - 10x + 5 \)

We need to find \( f\left(\frac{5}{4}\right) \).

### Step 1: Set up the equation for \( g(f(x)) \)

From the definition of \( g(x) \), we can express \( g(f(x)) \) as follows:

\[ g(f(x)) = f(x)^2 + f(x) - 1 \]

### Step 2: Equate the two expressions for \( g(f(x)) \)

We know that:

\[ g(f(x)) = 4x^2 - 10x + 5 \]

Thus, we can set the two expressions equal to each other:

\[ f(x)^2 + f(x) - 1 = 4x^2 - 10x + 5 \]

### Step 3: Rearrange the equation

Rearranging the equation gives us:

\[ f(x)^2 + f(x) - (4x^2 - 10x + 6) = 0 \]

This is a quadratic equation in terms of \( f(x) \):

\[ f(x)^2 + f(x) - 4x^2 + 10x - 6 = 0 \]

### Step 4: Use the quadratic formula

We can solve for \( f(x) \) using the quadratic formula:

\[ f(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = 1 \), and \( c = -(4x^2 - 10x + 6) \).

Calculating the discriminant:

\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-(4x^2 - 10x + 6)) = 1 + 16x^2 - 40x + 24 \]
\[ = 16x^2 - 40x + 25 \]

### Step 5: Substitute back into the quadratic formula

Now substituting back into the formula:

\[ f(x) = \frac{-1 \pm \sqrt{16x^2 - 40x + 25}}{2} \]

### Step 6: Evaluate \( f\left(\frac{5}{4}\right) \)

Now we need to find \( f\left(\frac{5}{4}\right) \):

First, substitute \( x = \frac{5}{4} \):

\[ f\left(\frac{5}{4}\right) = \frac{-1 \pm \sqrt{16\left(\frac{5}{4}\right)^2 - 40\left(\frac{5}{4}\right) + 25}}{2} \]

Calculating \( 16\left(\frac{5}{4}\right)^2 \):

\[ 16 \cdot \frac{25}{16} = 25 \]

Calculating \( -40\left(\frac{5}{4}\right) \):

\[ -40 \cdot \frac{5}{4} = -50 \]

Now substituting these values into the discriminant:

\[ f\left(\frac{5}{4}\right) = \frac{-1 \pm \sqrt{25 - 50 + 25}}{2} \]
\[ = \frac{-1 \pm \sqrt{0}}{2} \]
\[ = \frac{-1}{2} \]

### Final Answer

Thus, we find:

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

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