Given \( h(x)=3^{-x} \) and \( k^{-1}(x)=2 x^{2} \) for \( x \geq 0 \). A(0,57;0,53) is a point of intersection between \( h \) and \( k \). 4.1 Write down the equation of \( k \) in the form \( y=\ldots \) 4.2 Sketch the graphs of \( h \) and \( k \) on the same set of axes, clearly indicating the intercepts with the axes. 4.3 Write down the range of \( h \). 4.4 For which values of \( x \) is \( k(x) \leq h(x) \) ? 4.5 For which values of \( t \) will \( k(x)+t=h(x) \) have no real roots?

To determine \( k(x) \), we need to find its inverse relation from \( k^{-1}(x) = 2x^2 \). Therefore, if we express \( y = k^{-1}(x) \), this gives us: \( k(x) = \frac{1}{2\sqrt{x}} \) for \( x \geq 0 \). --- To solve for \( k(x) + t = h(x) \), we want the equation to have no real roots. For the function to have no intersection with the graph of \( h(x) = 3^{-x} \), \( k(x) + t \) must be greater than \( h(x) \) for all \( x \). This happens when \( t \) exceeds the maximum of \( h(x) \) since \( h(x) \to 0 \) as \( x \to \infty \). Thus, we set \( t > 3^{-0} - k(0) = 1 - 0 \). Therefore, \( t > 1 \) will yield no real roots for the equation \( k(x) + t = h(x) \). --- For further reading, you might explore the fascinating world of exponential functions and their inverses. Understanding these concepts will not only brighten your mathematical prowess but also reveal their numerous applications in real life, such as population growth models, radioactive decay, and in finance for calculating compound interest. On another note, diving deeper into the properties of functions can illuminate behaviors like monotonicity, which is essential in analyzing points of intersection. The study of calculus provides powerful tools, like derivatives, to help you understand the steepness of curves and helps you avoid common pitfalls in function analysis!