Point (3,2) lies on the graph of the inverse of `f(x)=2x^3+x+A`. The value of A is

Options

A. -54
B. -15
C. 15
D. 18

Explanation

To find the value of \( A \) such that the point \( (3, 2) \) lies on the graph of the inverse of the function \( f(x) = 2x^3 + x + A \), we can follow these steps:

### Step 1: Write the function in terms of \( y \)
We start with the function:
\[ f(x) = 2x^3 + x + A \]
We can express this as:
\[ y = 2x^3 + x + A \]

### Step 2: Swap \( x \) and \( y \)
To find the inverse function, we swap \( x \) and \( y \):
\[ x = 2y^3 + y + A \]

### Step 3: Substitute the point \( (3, 2) \)
Since the point \( (3, 2) \) lies on the graph of the inverse, we can substitute \( x = 3 \) and \( y = 2 \) into the equation:
\[ 3 = 2(2)^3 + 2 + A \]

### Step 4: Calculate \( 2(2)^3 + 2 \)
Now, we calculate \( 2(2)^3 + 2 \):
\[ 2(2)^3 = 2 \times 8 = 16 \]
Adding \( 2 \):
\[ 16 + 2 = 18 \]

### Step 5: Set up the equation
Now we have:
\[ 3 = 18 + A \]

### Step 6: Solve for \( A \)
To find \( A \), we rearrange the equation:
\[ A = 3 - 18 \]
\[ A = -15 \]

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{-15} \]

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