Question

A function \( f : \mathbb{R} \to A \) defined as \( f(x) = x^2 + 1 \) is onto, if \( A \) is: {5pt}

• A. \( (-\infty, \infty) \)
• B. \( (1, \infty) \)
• C. \( [1, \infty) \)
• D. \( [-1, \infty) \)
  {5pt}

Hint:

For quadratic functions \( f(x) = ax^2 + bx + c \): - The range depends on the vertex of the parabola. - For \( a>0 \), the range starts from the minimum value.

Answer 1

The Correct Option is C

Step 1: Function Definition.
The function is given by \( f(x) = x^2 + 1 \), with the domain \( x \in \mathbb{R} \). Since \( x^2 \ge 0 \) for all real \( x \), \( f(x) \) will be at least 1. 
Step 2: Determining the Range. 
The minimum value of \( x^2 \) is 0. Therefore, the minimum value of \( f(x) \) is: \[ f(x) = 0 + 1 = 1. \] The range of \( f(x) \) is thus \( [1, \infty) \). 
Step 3: Onto Condition. 
For \( f(x) \) to be an onto function, its codomain \( A \) must be equal to its range. Hence, \( A \) must be \( [1, \infty) \). 
Step 4: Final Statement. 
The function \( f(x) = x^2 + 1 \) is onto if and only if the codomain \( A \) is \( [1, \infty) \). {10pt}