Question

which of the following accurately describes the transformation from $f(x)=\cos(x)$ and $g(x)=\cos(-x)$?
statement #1: the function $g(x)$ is a reflection of $f(x)$ across the $y$-axis.
statement #2: the function $g(x)$ is a reflection of $f(x)$ across the $x$-axis.
statement #3: the function $g(x)$ is a reflection of $f(x)$ across $y = x$.
(1 point)
statement # is correct.
check answer remaining attempts : 3

Explanation:

Step1: Recall the property of cosine function

We know that \(\cos(-x)=\cos(x)\).

Step2: Analyze the reflection across axes

• Reflection across the \(y\) - axis: For a function \(y = f(x)\), its reflection across the \(y\) - axis is \(y=f(-x)\). Since \(g(x)=\cos(-x)\) and \(f(x)=\cos(x)\), and \(\cos(-x)=\cos(x)\), \(g(x)\) is the reflection of \(f(x)\) across the \(y\) - axis.
• Reflection across the \(x\) - axis: For a function \(y = f(x)\), its reflection across the \(x\) - axis is \(y=-f(x)\). Here, \(g(x)

eq - f(x)\) (because \(\cos(-x)
eq-\cos(x)\) in general).

• Reflection across \(y = x\): For a function \(y = f(x)\), its reflection across \(y=x\) is \(x = f(y)\). The form of \(g(x)=\cos(-x)\) is not in the form of reflection of \(y = \cos(x)\) across \(y=x\).

Answer:

Statement #1 is correct.