91Ó°ÊÓ

Inverse trigonometric functions are functions that reverse the action of standard trigonometric functions. In our exercise, we're dealing with the inverse tangent function, denoted as \( \tan^{-1} \). This function gives us the angle whose tangent is a given number.\
The range of \( \tan^{-1} \) is a set of outputs that the function can produce from its inputs. For positive inputs, such as \( t \) where \( t = x^2 + x + k \) can be any non-negative number, \( \tan^{-1}(t) \) will output angles ranging from \( 0 \) to \( \frac{\pi}{2} \) (radians).\
When we want the function \( f(x) = \tan^{-1}(x^2 + x + k) \) to be onto, it means the output must span every possible angle in \( [0, \frac{\pi}{2}) \). This requires \( x^2 + x + k \) to cover the entire interval \([0, \infty)\) so that the inverse tangent can produce all the angles in the range of \( f \). This leads us to consider the behavior of the quadratic expression.

A quadratic expression is a polynomial of degree two, typically in the form \( ax^2 + bx + c \). In our exercise, the quadratic component is \( x^2 + x + k \). Understanding the range of values this can produce is crucial for determining the behavior of our function.\
The general strategy for analyzing a quadratic expression is to find its minimum (or maximum) value. The vertex form or using the vertex formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients of \( x^2 \) and \( x \), helps determine the turning point. In this case, \( a = 1 \) and \( b = 1 \), so the minimum occurs at \( x = -\frac{1}{2} \).\
Evaluating \( g(x) \) at this point gives \( g(-\frac{1}{2}) = \frac{1}{4} - \frac{1}{2} + k = -\frac{1}{4} + k \). For the quadratic to cover \([0, \infty)\), we need this minimum to be at least zero, implying \( k \geq \frac{1}{4} \). Thus, \( k = \frac{1}{4} \) is the smallest constant that enables our function to be onto.

The range of a function refers to all the possible outputs it can produce based on its inputs. For our function \( f(x) = \tan^{-1}(x^2 + x + k) \), determining the range helps establish whether \( f \) is onto, meaning the entire interval \([0, \frac{\pi}{2})\) is covered.\
The inverse tangent function inherently produces outputs in this range when given inputs from \([0, \infty)\). Thus, by ensuring the quadratic expression \( x^2 + x + k \) spans \([0, \infty)\), \( f(x) \) will map every \( x \) in \( \mathbb{R} \) to every \( y \) in \( A \).\
This linkage between the input and output intervals translates into setting the minimum value of \( g(x) = x^2 + x + k \) to \( 0 \) or higher. This ensures that every real number has a corresponding output in \( A \), fulfilling the criteria for \( f(x) \) to be onto and confirming \( k = \frac{1}{4} \) as pivotal for achieving full coverage.