Enrichment, Extension, and Application

1 point

Exercises

1. On the same graph, sketch the graphs of $f(x) = \frac{1}{1-x}$ and $g(x) = \frac{1}{x-1}$
2. Let $$\varepsilon$$ be any positive real number. Find the values of the following.
1. $$f(1+\varepsilon)$$
2. $$f(1-\varepsilon)$$
3. $$g(1+\varepsilon)$$
4. $$g(1-\varepsilon)$$
3. Sketch the graph of $$h(x) = f(x) + g(x)$$. What is the value of $$h(1+\varepsilon)$$ and $$h(1-\varepsilon)$$ for any value of $$\varepsilon \gt 0$$?
1. On September  1$$^{st}$$ of each year from  2000 – 2014 (inclusive), I invested $$P$$ to help pay for the costs of my child's education. Assume that the interest rate earned on this money is $$100i~\%$$ per year and that it remains unchanged over the entire $$15$$ year period.

Let $$f_{15}(i)$$ be a function giving the total value of the accumulated funds after $$15$$ years of investing the money (that is, from September 1, 2000 to August 31, 2014 ) at a given interest rate of $$100i~\%$$, where $$i \gt 0$$.

1. Show that $$f_{15}(i)$$ can be written as a rational function.
2. Determine $$f_{15}(0.03)$$, if $$P = 1000$$.
3. If my child takes the money in $$4$$ equal installments at the beginning of each year while investing the remainder at $$100i\%$$, how much money can be withdrawn each year? (Assume $$i = 0.03$$).
2. If $\frac{(x-2013)(y-2014)}{(x-2013)^2+(y-2014)^2} = \frac{1}{2}$ determine the value of $$x+y$$.
3. Consider the function $f(x) =\frac{(x+k)^2}{x^2+1}$ where $$k$$ is a constant and $$k \gt 0$$. Prove that $$f(x) \le k^2+1$$.
1. Determine the $$x$$-intercepts, $$y$$-intercepts, and all asymptotes for the function $f(x) = \frac{x+4}{2x-5}$
2. Find all values of $$x$$ such that $f(x) = \frac{x+4}{2x-5} \ge 0$
1. If $$T = x^2 + \dfrac{1}{x^2}$$, determine the values of $$b$$ and $$c$$ so that $$\dfrac{x^6+1}{x^6} = T^3+bT+c$$ for all non-zero real numbers $$x$$.
2. If $$x$$ is a real number satisfying $$x^3+\dfrac{1}{x^3} = 2\sqrt{5}$$, determine the exact value of $$x^2 + \dfrac{1}{x^2}$$.
4. If $f(x) = \frac{3x-7}{x+1}$ and $$g(x)$$ is the inverse of $$f(x)$$, find the value of $$g(2)$$.
5. For any positive integer $$n$$, define $f(n) = \frac{1}{2\sqrt{n}}$ Prove that $$f(n+1) \lt \sqrt{n+1}-\sqrt{n} \lt f(n)$$.
6. The function $f(x)=\frac{a+3x}{(x-1)(x+1)}$ is defined for all real $$x, x \ne \pm 1$$. Find all values of $$a$$ such that for any one of those values of $$a$$, the range of $$f(x)$$ is the set of real numbers.
7. Define $$f(x) = \left \lfloor x \right \rfloor$$ to be the floor function for the real number $$x$$. (If $$f(x)$$ is the floor function, then $$f(x)$$ is the largest integer less than or equal to $$x$$; for example, $$f(3.41) = 3$$ (see the Enrichment Problems for Unit 1)).

Find all positive values of $$x$$ such that $\frac{1}{f(x)}+\frac{1}{f(3x)} = x-f(x)$