Problem 79 - College Algebra and Calculus: An Applied Approach

Question

If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)

Step-by-Step Solution

Answer

(a) The function \(g(x) = -f(x)\) is odd. (b) The function \(g(x) = f(-x)\) is even. (c) The function \(g(x) = f(x) - 2\) is even. (d) The function \(g(x) = f(x - 2)\) is neither even nor odd.

Step 1: Determine Nature of \(g(x) = -f(x)\)

By substituting \(-x\) for \(x\) in \(g(x)\), we get \(g(-x) = -f(-x)\). Because \(f\) is an even function, \(f(-x) = f(x)\). Therefore, \(g(-x) = -f(x)\), which is the opposite of \(g(x)\). So, \(g(x)\) is an odd function.

Step 2: Determine Nature of \(g(x) = f(-x)\)

Substitute \(-x\) for \(x\) in \(g(x)\) to get \(g(-x) = f(x)\). This is exactly equal to \(g(x)\). Hence, \(g(x)\) is an even function.

Step 3: Determine Nature of \(g(x) = f(x) - 2\)

Substitute \(-x\) for \(x\) in \(g(x)\) to get \(g(-x) = f(-x) -2\). Since \(f\) is an even function, \(f(-x) = f(x)\). So, \(g(-x) = f(x) - 2\), which is exactly equal to \(g(x)\). Hence, \(g(x)\) is an even function.

Step 4: Determine Nature of \(g(x) = f(x - 2)\)

Substitute \(-x\) for \(x\) in \(g(x)\) to get \(g(-x) = f(-x - 2)\). This does not simplify to \(g(x)\) or \(-g(x)\). So, the resulting function \(g(x)\) is neither even nor odd.