Question

If \( f(x) = 2x^2 + bx + c \), \( f(0) = 3 \) and \( f(2) = 1 \), then \( (f \circ f)(1) = \)

• A. 0
• B. 2
• C. 1
• D. 3

Hint:

In problems involving function composition, first calculate the inner function value and then substitute it into the outer function.

Answer 1

The Correct Option is C

Step 1: Use the given conditions.
We are given \( f(0) = 3 \) and \( f(2) = 1 \), so we can substitute these into the equation \( f(x) = 2x^2 + bx + c \) to find the values of \( b \) and \( c \). \[ f(0) = 2(0)^2 + b(0) + c = 3 \quad \Rightarrow \quad c = 3 \] \[ f(2) = 2(2)^2 + b(2) + 3 = 1 \quad \Rightarrow \quad 8 + 2b + 3 = 1 \quad \Rightarrow \quad 2b = -10 \quad \Rightarrow \quad b = -5 \]
Step 2: Find \( (f \circ f)(1) \).
Now that we know \( f(x) = 2x^2 - 5x + 3 \), we calculate \( f(1) \): \[ f(1) = 2(1)^2 - 5(1) + 3 = 2 - 5 + 3 = 0 \] Now, calculate \( (f \circ f)(1) = f(f(1)) = f(0) \): \[ f(0) = 3 \]
Step 3: Conclusion.
The correct answer is 1.