Relations & Functions
Function Properties
Grade 12
Question:
<p>Let <span class='math'>f : \mathbb{R} \to \mathbb{R}</span> and <span class='math'>g : \mathbb{R} \to \mathbb{R}</span> be two one-one and onto functions, such that they are the mirror images of each other about the line <span class='math'>y = a</span>. If <span class='math'>h(x) = f(x) + g(x)</span>, then <span class='math'>h(x)</span> is</p>
<p>(a) one-one and onto</p>
<p>(b) only one-one and not onto</p>
<p>(c) only onto but not one-one</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: When two functions are mirror images about a horizontal line, their sum is constant, making h(x) a constant function.
<p>Since <span class='math'>f</span> and <span class='math'>g</span> are mirror images about <span class='math'>y = a</span>, we have <span class='math'>f(x) + g(x) = 2a</span> for all <span class='math'>x \in \mathbb{R}</span>. Therefore, <span class='math'>h(x) = 2a</span> is a constant function. A constant function is onto but not one-one.</p>
Correct Answer: c