If `f(x) = 2 x^3 + 7x - 5` then `f^(-1) (4)` is :

Options

A. 1
B. 2
C. `1//3`
D. non-existent

Answer

To find \( f^{-1}(4) \) for the function \( f(x) = 2x^3 + 7x - 5 \), we will follow these steps:

### Step 1: Set up the equation
We start by letting \( y = f(x) \). This gives us the equation:
\[ y = 2x^3 + 7x - 5 \]
To find \( f^{-1}(4) \), we set \( y = 4 \):
\[ 4 = 2x^3 + 7x - 5 \]

### Step 2: Rearrange the equation
Rearranging the equation, we get:
\[ 2x^3 + 7x - 5 - 4 = 0 \]
This simplifies to:
\[ 2x^3 + 7x - 9 = 0 \]

### Step 3: Let \( x = T \)
To simplify our calculations, we let \( T = x \). Thus, we rewrite the equation as:
\[ 2T^3 + 7T - 9 = 0 \]

### Step 4: Factor the equation
We will try to find rational roots using the Rational Root Theorem. Testing \( T = 1 \):
\[ 2(1)^3 + 7(1) - 9 = 2 + 7 - 9 = 0 \]
Since \( T = 1 \) is a root, we can factor the polynomial as:
\[ (T - 1)(2T^2 + 2T + 9) = 0 \]

### Step 5: Solve the quadratic equation
Now, we need to solve the quadratic equation \( 2T^2 + 2T + 9 = 0 \) using the discriminant:
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 2 \cdot 9 = 4 - 72 = -68 \]
Since the discriminant is negative, there are no real roots from this quadratic equation.

### Step 6: Conclusion
The only real solution we have is from \( T - 1 = 0 \), which gives us:
\[ T = 1 \]
Thus, we conclude that:
\[ f^{-1}(4) = 1 \]

### Summary
The value of \( f^{-1}(4) \) is \( 1 \).

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