Question

Describe how the functions \(f(x)=3+\llbracket x \rrbracket\) and \(g(x)=3-\llbracket-x \rrbracket\) differ.

Short answer

The function \( f(x) = 3 + \llbracket x \rrbracket \) increases steadily for positive x values and decreases for negative x ones, following the base floor function in the positive x-axis. On the contrary, \( g(x) = 3 - \llbracket -x \rrbracket \) decreases for positive x values and increases for negative ones, following the base floor function in the negative x-axis. Therefore, \( f(x) \) and \( g(x) \) largely exhibit contrasting behavior depending on the sign of \( x \).

Step-by-step solution

Understand the behaviour of floor functions

The floor function, also known as the greatest integer function, is given by \(\llbracket x \rrbracket\), where \(\llbracket x \rrbracket\) is the largest integer less than or equal to x. For example, \(\llbracket 3.7 \rrbracket = 3\) , \(\llbracket -2.3 \rrbracket = -3\). The floor function steps down to the next lowest integer, representing a step function that forms a staircase pattern.

Analyze function \( f(x)=3+\llbracket x \rrbracket \)

Notes that for positive \(x\) values, the floor function of \(x\) would simply be \(x\) rounded down to the nearest lower integer, hence the function \( f(x) \) increases steadily for positive \(x\). However, for negative \(x\) values, the floor function of \(x\) would be \(x\) rounded down to the nearest lower integer which is actually higher in absolute value, hence \( f(x) \) decreases steadily for negative \(x\). This function will closer reflect the base floor function graph.

Analyze function \( g(x)=3-\llbracket -x \rrbracket \)

In this function, there's a negative sign in front of the \(x\) within the floor function, making it \(-x\). This means for positive \(x\) values, the floor function would return the opposite of \(x\) rounded down. Hence, \( g(x) \) decreases steadily for positive \(x\). For negative 'x' values, the floor function returns the negative of \(x\) rounded down which is actually less in absolute value. Hence, \( g(x) \) increases steadily for negative \(x\). This reflects a flipped version of the base floor function.

Comparing the two functions

Contrasting \( f(x) \) and \( g(x) \), it can be said that they generally oppose the floor function's base pattern in different areas of the x-axis. \( f(x) \) follows the step-up pattern of the base floor function in the positive x-axis whereas \( g(x) \) follows the step-up pattern in the negative x-axis. In other quadrants, each shows a contrasting 'step down' pattern.