Question

Let \( f(x) = x^2 - 10x - 19, \, x \in \mathbb{R} \). Then the inverse image of 5, \( f^{-1}(5) = \)

• A. \( \{-2, -12\} \)
• B. \( \{-2, 12\} \)
• C. \( \{2, -12\} \)
• D. \( \{2, 12\} \)
• E. \( \phi \)

Hint:

Inverse image means solving \( f(x)=k \), not finding inverse function.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The inverse image of a value `y` under a function `f(x)`, denoted as f\(^{-1}\)(y), is the set of all `x` values in the domain of `f` such that `f(x) = y`. It is important not to confuse the inverse image with the inverse function.
Step 2: Key Formula or Approach:
To find the inverse image of 5, we need to solve the equation f(x) = 5.
\[ x^2 - 10x - 19 = 5 \] Step 3: Detailed Explanation:
First, set up the equation:
\[ x^2 - 10x - 19 = 5 \] Rearrange the equation to form a standard quadratic equation (ax\(^2\) + bx + c = 0):
\[ x^2 - 10x - 19 - 5 = 0 \] \[ x^2 - 10x - 24 = 0 \] Now, solve this quadratic equation. We can factor it by finding two numbers that multiply to -24 and add to -10. These numbers are -12 and 2.
\[ (x - 12)(x + 2) = 0 \] This gives two possible values for x:
\[ x - 12 = 0 \quad \text{or} \quad x + 2 = 0 \] \[ x = 12 \quad \text{or} \quad x = -2 \] The set of these x-values is the inverse image of 5.
Step 4: Final Answer:
The inverse image of 5, f\(^{-1}\)(5), is the set \{-2, 12\}.