# Enrichment, Extension, and Application

1 point

## Exercises

1. The function $$f(x)$$ satisfies the equation $$f(x) = f(x - 1) + f(x + 1)$$ for all values of $$x$$. Define $$f(1) = 1$$ and $$f(3) = 3$$; then, $$f(2) = 1 + 3 = 4$$. Determine the value of $$f(1867)$$.
1. For $$x$$ and $$k$$ real numbers, determine the values of of $$k$$ for which the graphs of $$f(x) = x^2-4$$ and $$g(x) = 2|x| + k$$ do not intersect.
2. For $$x$$ and $$k$$ real numbers, determine the values of of $$k$$ for which the graphs of $$f(x) = x^2-4$$ and $$g(x) = 2|x| + k$$ intersect in exactly two points.
2. For $$x$$ and $$k$$ real numbers, for what values of $$k$$ will the graphs of $$f(x) = -2\sqrt{x+1}$$ and $$g(x) = \sqrt{x-2}+k$$ intersect?
1. The equation $$y = x^2 + 2ax + a$$ represents a parabola for all real values of $$a$$. Prove that each of these parabolas passes through a common point and determine the coordinates of this point.
2. The vertices of the parabolas in part a) above lie on a curve. Prove that this curve is itself a parabola whose vertex is the common point found in part a).
1. A fixed point of a function is an element of the domain that is mapped to itself by the function. That is, if $$f(x)$$ is a function and $$f(p) = p$$, then $$p$$ is a fixed point of the function $$f(x)$$. Determine the fixed points, if any, of the following functions (assume $$x$$ is an element of the real numbers):
1. $$f(x) = 4x^2 - x - 6$$
2. $$f(x) = \dfrac{x}{1-x}$$
3. $$f(x) = 2^{x-1}$$
4. $$f(x) = \left\lvert x \right\rvert - 2$$
2. For what values of $$k$$ will $$f(x) = x^2 + kx + 4$$ have no fixed points, exactly one fixed point, and exactly two fixed points?
3. Let $$f(x) = ax^2 + bx + c$$, where $$a, b,$$ and $$c$$ are real numbers, be a function defined on the real numbers. If $$2f(x) = f(2x)$$ for all values of $$x$$, determine the values of $$a$$, $$b$$, and $$c$$.
4. Floor and Ceiling Functions:

Define the floor for a real number, $$x$$, to be the the largest integer $$\le x$$. We write the floor of $$x$$ as $$\lfloor x \rfloor$$.

Define the ceiling for a real number, $$x$$, to be the smallest integer $$\ge x$$. We write the ceiling of $$x$$ as $$\lceil x \rceil$$.

1. If $$f(x) = \lfloor x \rfloor$$, find $$f(8.6), f(2), f(-2.6)$$.
2. If $$g(x) = \lceil x \rceil$$, find $$g(8.6), g(2), g(-2.6)$$
3. On the same axes, draw the graphs of $$f(x) = \lfloor x \rfloor$$, and $$g(x) = \lceil x \rceil$$, for $$-5 \leq x \leq 5$$.
4. Draw the graph of $$g(x) - f(x)$$ for $$-5 \leq x \leq 5$$.
5. Rounding Function To round a decimal expression to $$k$$ decimal places, we add $$5 \times 10^{-(k+1)}$$ to the number and then keep the first $$k$$ numbers after the decimal point. For example, $$5.61497$$ rounded to three decimal places becomes $$5.615$$ (since $$5.61497 + 0.0005 = 5.61547$$, which becomes $$5.615$$ when rounded to the first three decimal places); $$8.297$$ rounded to two decimal places becomes $$8.30$$; $$12.34712$$ rounded to four decimal places becomes $$12.3471$$. If $$r_k(x)$$ is a function that takes the real number $$x$$ and rounds it to $$k$$ decimal places, give an expression for $$r_k(x)$$ in terms of $$x$$ and $$k$$. (Hint: Use the floor function defined above).
6. Indicator Function An Indicator function, $$f(x)$$, is a function that takes the value $$1$$ if some condition on $$x$$ is true, and the value $$0$$ if the condition is not true. For integers $$x$$, define an indicator function: $f(x) = \begin{cases} 1 & \text{if $$x$$ is divisible by $$2$$} \\ 0 & \text{if $$x$$ is not divisible by $$2$$} \end{cases}$ and define another indicator function: $g(x) = \begin{cases} 1 & \text{if $$x$$ is divisible by $$3$$} \\ 0 & \text{if $$x$$ is not divisible by $$3$$} \end{cases}$
1. Find $$f(2), f(9), f(4086), g(2), g(9), g(4086)$$.
2. Let $$h(x) = f(x)g(x)$$. Determine whether $$h(x)$$ is an indicator function, and, if so, describe in words the condition it indicates.
3. Let $$k(x) = \left \lfloor \dfrac{f(x) + g(x) + 1}{2} \right \rfloor$$. Determine whether $$k(x)$$ is an indicator function, and, if so, describe in words the condition it indicates.
7. Modulo Function The function $$m_k(x)$$ has as its domain the non-negative integers. It is defined such that $$m_k(x)$$ is the smallest non-negative remainder when $$x$$ is divided by the positive integer $$k$$. For example, $$m_4(14) = 2; m_{13}(42) = 3$$.
1. Find $$m_9(101), m_6(720), m_7(55)$$.
2. Give the range of the function $$m_k(x)$$.
3. Give a mathematical expression for $$m_k(x)$$ in terms of $$k$$ and $$x$$. (Hint: Use the Floor Function defined above).
4. Draw the graph of the function $$m_4(x)$$, for $$0 \le x \le 15$$.
8. Prime Counting Function Define the function $$\pi(x)$$ to be the number of prime numbers less than or equal to $$x$$, where $$x$$ is a real number. Draw a graph of $$\pi(x)$$ for $$0 \lt x \leq 50$$.