Enrichment, Extension, and Application

Question 1

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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equal installments at the beginning of each year while investing the remainder at \( 100i\% \), how much money can be withdrawn each year?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 \[\frac{1}{f(x)}+\frac{1}{f(3x)} = x-f(x)\]

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