Let `g(x) = x^(2) + x -1 AA x in R` and `(gof)(x) = 4x^(2) + 10 x + 5 AA x in R`, then `f(7//2)` = …………

Answer

To solve the problem step by step, we need to find the value of \( f\left(\frac{7}{2}\right) \) given the functions \( g(x) = x^2 + x - 1 \) and \( (g \circ f)(x) = 4x^2 + 10x + 5 \).

### Step 1: Understand the composition of functions
We know that \( (g \circ f)(x) = g(f(x)) \). This means we will substitute \( f(x) \) into the function \( g(x) \).

### Step 2: Write out the expression for \( g(f(x)) \)
Given \( g(x) = x^2 + x - 1 \), we can express \( g(f(x)) \) as:
\[ g(f(x)) = (f(x))^2 + f(x) - 1 \]

### Step 3: Set up the equation
We know that \( g(f(x)) = 4x^2 + 10x + 5 \). Therefore, we can equate:
\[ (f(x))^2 + f(x) - 1 = 4x^2 + 10x + 5 \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ (f(x))^2 + f(x) - 4x^2 - 10x - 6 = 0 \]

### Step 5: Assume a linear form for \( f(x) \)
Assume \( f(x) \) is a linear function of the form \( f(x) = ax + b \). We will substitute this into our equation.

### Step 6: Substitute \( f(x) \) into the equation
Substituting \( f(x) = ax + b \) into the equation:
\[ (ax + b)^2 + (ax + b) - 4x^2 - 10x - 6 = 0 \]
Expanding \( (ax + b)^2 \) gives:
\[ a^2x^2 + 2abx + b^2 + ax + b - 4x^2 - 10x - 6 = 0 \]
This simplifies to:
\[ (a^2 - 4)x^2 + (2ab + a - 10)x + (b^2 + b - 6) = 0 \]

### Step 7: Set coefficients equal to zero
For the equation to hold for all \( x \), the coefficients must be equal to zero:
1. \( a^2 - 4 = 0 \)
2. \( 2ab + a - 10 = 0 \)
3. \( b^2 + b - 6 = 0 \)

### Step 8: Solve for \( a \)
From \( a^2 - 4 = 0 \):
\[ a^2 = 4 \implies a = 2 \text{ or } a = -2 \]

### Step 9: Solve for \( b \)
1. If \( a = 2 \):
\[ 2(2)b + 2 - 10 = 0 \implies 4b - 8 = 0 \implies b = 2 \]
2. If \( a = -2 \):
\[ 2(-2)b - 2 - 10 = 0 \implies -4b - 12 = 0 \implies b = -3 \]

### Step 10: Possible functions for \( f(x) \)
Thus, we have two possible functions:
1. \( f(x) = 2x + 2 \)
2. \( f(x) = -2x - 3 \)

### Step 11: Calculate \( f\left(\frac{7}{2}\right) \)
1. For \( f(x) = 2x + 2 \):
\[ f\left(\frac{7}{2}\right) = 2\left(\frac{7}{2}\right) + 2 = 7 + 2 = 9 \]
2. For \( f(x) = -2x - 3 \):
\[ f\left(\frac{7}{2}\right) = -2\left(\frac{7}{2}\right) - 3 = -7 - 3 = -10 \]

### Final Answer
Thus, the possible values for \( f\left(\frac{7}{2}\right) \) are \( 9 \) and \( -10 \).

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