Verified Answer :

  • Identify the transformation rule for reflection over the y-axis, which is \(g(x) = f(-x)\).
  • Substitute \(-x\) into the function \(f(x) = x^2 - 1\) to get \(g(x) = (-x)^2 - 1\).
  • Simplify the expression to \(g(x) = x^2 - 1\).
  • Match the result with option B, which is \(\boxed{B}\).

Explanation

  1. Analyzing the Transformation Rule

The problem asks us to find the equation for a new function \(g(x)\) which is the reflection of the original function \(f(x) = x^2 - 1\) over the y-axis. The reflection of a function \(f(x)\) over the y-axis is obtained by performing the transformation \(f(-x)\). This means that for every instance of \(x\) in the original equation, we substitute it with \(-x\).

  1. Performing the Transformation

To find \(g(x)\), we substitute \(-x\) into the original function \(f(x) = x^2 - 1\). This gives us:
\[g(x) = f(-x) = (-x)^2 - 1\]
Recall that any real number squared becomes positive, regardless of its original sign. Therefore, \((-x)^2 = x^2\). Substituting this back into our expression, we get:
\[g(x) = x^2 - 1\]

  1. Comparing Results with Options

Comparing our result \(g(x) = x^2 - 1\) to the provided options:

A\(g(x) = -x^2 - 2\)
B\(g(x) = x^2 - 1\)
C\(g(x) = x^2 + 1\)

The equation \(g(x) = x^2 - 1\) matches option B. This indicates that reflecting the parabola \(f(x) = x^2 - 1\) over the y-axis does not change the function, because the parabola is already symmetric about the y-axis. The final answer is B.

Examples

Reflecting functions over axes is a fundamental concept in coordinate geometry. For example, if you are designing a logo that is symmetrical across a central vertical line, you can create one half of the design as a function \(f(x)\) and reflect it over the y-axis to complete the other side of the logo. This ensures perfect symmetry in your graphic design. In our case, because the function \(f(x) = x^2 - 1\) is already symmetric, the reflection results in the same shape.

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