e. For what value of \( a \) does the line with equation \( y=x \) intersect the graph of \( y=f(x) \) at the point with coordinates \( (c, c) \), where \( c \) is a positive real number?

To find the value of \( a \) for which the line \( y = x \) intersects the graph of \( y = f(x) \) at the point \( (c, c) \), we need to ensure that both equations yield the same \( y \)-value when \( x = c \). Thus, the equation can be set up as \( f(c) = c \). This means that for the intersection to occur at a point \( (c, c) \), the function \( f(x) \) must pass through the point where \( x \) and \( f(x) \) are equal. Beyond finding just that intersection, it will be integral to explore how the specifics of \( f(x) \) express on either side of that point for all values of \( a \). To get a more comprehensive view of \( f(x) \), one might consider experimenting with various functional types, such as linear functions with slopes that differ from 1, or even quadratic polynomials. Every decision on the nature of \( f(x) \) can dramatically shift where \( a \) aligns to maintain that intersection at \( (c, c) \).