Relations & Functions
Properties of Even and Odd Functions
Grade 12
Question:
<p>The functions <strong>f₁(x) = log(x + √(x² + 1))</strong> and <strong>f₂(x) = x · ((aˣ - 1)/(aˣ + 1))</strong> are respectively:</p>
<p>(a) odd and even function</p>
<p>(b) even and odd function</p>
<p>(c) odd and odd function</p>
<p>(d) even and even function</p>
Step-by-Step Solution
Key Concept: For odd functions: f(-x) = -f(x); For even functions: f(-x) = f(x). Check both properties for given functions.
<p><strong>For f₁(x) = log(x + √(x² + 1)):</strong></p><p>f₁(-x) = log(-x + √(x² + 1)) = log(1/(x + √(x² + 1))) = -log(x + √(x² + 1)) = -f₁(x)</p><p>Therefore, f₁(x) is an odd function.</p><p><strong>For f₂(x) = x · ((aˣ - 1)/(aˣ + 1)):</strong></p><p>f₂(-x) = (-x) · ((a⁻ˣ - 1)/(a⁻ˣ + 1)) = (-x) · ((1 - aˣ)/(1 + aˣ)) = (-x) · (-(aˣ - 1)/(aˣ + 1)) = x · ((aˣ - 1)/(aˣ + 1)) = f₂(x)</p><p>Therefore, f₂(x) is an even function.</p><p>∴ Answer is (a) odd and even function.</p>
Correct Answer: a