Relations & Functions
Range of a Function
Grade 12
Question:
<p>Given <em>f</em> : <em>R</em> → <em>R</em> be defined by \( f(x) = \dfrac{x}{1+x^2},\ x \in R \). Then <em>f</em>(<em>x</em>) ∈</p>
<p>\(\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]\)</p>
<p>\(\left(-\dfrac{1}{2}, \dfrac{1}{2}\right)\)</p>
<p>\(\left(0, \dfrac{1}{2}\right)\)</p>
<p>\(\mathbb{R}\)</p>
Step-by-Step Solution
Key Concept: Find the range of f(x) = x/(1+x²) by analyzing its critical points and behavior. The function has a maximum and minimum value that bound all possible outputs.
<p><strong>Step 1:</strong> Let y = f(x) = x/(1+x²). Rearrange to find bounds on y.</p><p><strong>Step 2:</strong> Multiply both sides: y(1+x²) = x, which gives yx² - x + y = 0</p><p><strong>Step 3:</strong> For real values of x to exist, the discriminant must be non-negative: Δ = 1 - 4y² ≥ 0</p><p><strong>Step 4:</strong> This gives 4y² ≤ 1, so y² ≤ 1/4, therefore -1/2 ≤ y ≤ 1/2</p><p><strong>Step 5:</strong> Verify endpoints: f(1) = 1/2 (maximum) and f(-1) = -1/2 (minimum)</p><p>∴ Answer: f(x) ∈ [-1/2, 1/2]</p>
Correct Answer: A