Exercises


Lesson 1: Introduction to Functions


  1. Consider the circles whose equations are given below. For each circle, find
    1. the coordinates of the centre.
    2. the radius.
    3. the coordinates of the endpoints of the vertical diameter.
    4. the coordinates of the endpoints of the horizontal diameter.
    1. \((x+1)^2+(y-3)^2=49\)
    2. \(x^2 + (y+5)^2=121\)
    3. \((x-7)^2+y^2=81\)
    1. A circle has diameter with endpoints \(A(8,2)\) and \(B(-6,2)\). Find the equation of this circle.
    2. Find the equation of the circle that has centre \(O(-1,1)\) and passes through \(P(3,-2)\).
  3. Determine a set of ordered pairs, each in terms of \(h\), \(k\), and \(r\), containing four points that are on the graph of the relation \((x-h)^2+(y-k)^2=r^2\).
  4. Determine, with justfication, whether each relation is a function.
    1. \(\{(1,2), ~(2,4), ~(5,6),~ (14, 7) \}\)
    2. x values map to y values. 2 maps to negative 3, 2 maps to 1, 3 maps to negative 3, and 4 maps to 5.
    3. A cubic relation with x-intercepts negative 4, negative 1, and 1. The function smoothly goes from quadrant 4 to quadrant 1.
    4. \(x^2+y^2=25\)
    5. \(y=(x-2)^2+3\)
  5. Given the mapping notation \(f:x \rightarrow2x^2+x+1\), determine the output, \(y\), when the input is \(x=-3\).
  6. Given the mapping notation \(g:x \rightarrow x^2+1\), determine the possible inputs, \(x\), when the output is \(y=10\).
  7. The area of a circle is given by \(A= \pi r^2\) where \(r\) is the radius.
    1. Is the relation that takes radius as the input and gives area as the output a function? Explain your answer.
    2. Determine the area of a circle that has a radius of \(4\) cm.
    3. Determine the radius of a circle that has area \(49 \pi \) m2.
  8. Two familiar units for measuring temperature are degrees Celsius (\(^\circ\)C) and degrees Fahrenheit (\(^\circ\)F). A third unit, perhaps less familiar but widely used in scientific applications, is Kelvin (K).

    Let \(g\) be the function that maps degrees Celsius to degrees Fahrenheit.

    Let \(h\) be the function that maps Kelvin to degrees Celsius.

    Then,

    • \(g: C \rightarrow \dfrac95 C + 32\), and
    • \(h: K \rightarrow K - 273.15\).

    Determine the mapping notation for the function, \(f\), that maps Kelvin to degrees Fahrenheit.


Lesson 2: Function Notation


  1. The graph shows the function \(f\).

    See adjacent alternative format.

    On the given domain, the function is composed of the following functions:

    Domain Function
    \(x \leq -3 \) \(y = 3x+12\)
    \( -3 \lt x \leq 0\) \(y = -x\)
    \(0 \lt x \leq 2\) \(y =x \)
    \(x \gt 2\) \(y =1\)
    1. Find the values of \(f(-2)\), \(f(2)\), and \(f(70)\).
    2. Find the value of \(x\) such that \(f(x)=3\).
    3. Find all possible values of \(x\) such that \(f(x)=0\).
  2. A function, \(h\), has mapping diagram as shown.

    x maps to h of x. 0 maps to negative 1, 1 maps to negative 5, 2 maps to 10, 3 maps to 2, and 4 maps to 11.

    1. Find \(h(4)\).
    2. Find \(h(1)+2h(0)\).
    3. Find \(h(h(3))\).
    4. Find a value of \(p\) such that \(h(p)=-5\).
  3. Given \(f(x)=3x+2\) and \(g(x)=-5x-1\), find the following.
    1. \(f(3)\)
    2. \(g(0)\)
    3. \(g(x+h)\)
    4. \(3f(2)-4g(-1)\)
    5. All \(x\) such that \(g(x)=3\)
  4. Given \(f(x)=3x^2-12\) and \(g(x)=x^2+4x+3\), find the following.
    1. \(2f(-2)-3g(0)\)
    2. \(f(a+2)\)
    3. \(g(x+h)\)
    4. \(x\), when \(g(x)=0\).
  5. Isha has developed a cellphone app that she hopes to sell. A website will initially charge her \($300\) to list the app for sale on the site, but will pay her \($1.50\) for each app that is sold.

    The profit that Isha will make can be modelled by the function

    \(P(x)=1.5x-300\)

    where \(x\) is the number of apps sold.

    1. How many apps must be sold so that Isha can recover the initial cost of \($300\)?
    2. If she sells \(1000\) apps, how much profit will she make?
    3. She is hoping to sell \(100\) apps per week. How many weeks will it take to earn a profit of \($5000\)?
  6. Determine the equation of the given linear functions.
    1. A line labelled y = f of x passing through the points A (negative 2, 5) and B (1,1).
    2. \(x\) \(f(x)\)
      \(0\) \(4\)
      \(5\) \(14\)
      \(10\) \(24\)
      \(15\) \(34\)
      \(20\) \(44\)
  7. Given \(f(x)=x^2+x\), find each of the following.
    1. \(f(a)\)
    2. \(f(b)\)
    3. \(f(a)+f(b)\)
    4. \(f(a+b)\)
    5. What conclusion can be drawn about \(f(a)+f(b)\) and \(f(a+b)\)?
  8. Determine an equation of the given quadratic functions.
    1. A parabola labelled y equals f of x that passes through the points (negative 3, 0), (one-half, 0) and (0, negative 6).
    2. \(x\) \(f(x)\)
      \(3-2\sqrt{5}\) \(0\)
      \(4\) \(5\)
      \(3+2\sqrt{5}\) \(0\)
    3. The quadratic function, \(f(x)\), passing through \((-4,7)\) and \((6,7)\) with one of the zeros at \(x=10\).
  9. The volume of water in a storage tank is given by \(V(t)=\dfrac13 t^2-6t+100\), where \(t\) is the time measured in \(\)hours. 
    1. Determine \(V(0)\) and \(V(24)\).
    2. Determine \(r \ne 0\) such that \(V(r)=V(0)\).
    3. Determine the time when the volume of the water in the tank is a minimum.
  10. Given \(f(x)=x\) and \(g(x)=x^2+bx+9\), find \(b\) such that \(y=f(x)\) and \(y=g(x)\) meet at exactly one point.

Lesson 3: Domain and Range


  1. Write the following using set notation.
    1. The set of all natural numbers that are greater than or equal to \(4\).
    2. The set of all real numbers that are greater than or equal to \(-3\) and less than \(7\).
  2. Determine the domain and range of the following functions.
    1. \(\{(-2,3),~ (3,4),~ (8,5),~ (12, 6)\}\)
    2. 1 maps to negative 3, 4 maps to 5, 9 maps to negative 4, 16 maps to 5, and 7 maps to 11.
    3. A graph with the following points plotted: (negative 2, 0), (negative 1, 3), (0, 1), (1, 3), and (2, 0).
    4. An open dot at (negative 3, 0). A line segment from (negative 3, 0) to (0, 3) inclusive. An open dot at (4, 0) connected to a line that passes through (4, 0).
    5. Part of a parabola that opens upwards and has end points (negative 2, 6) and (4, 0) with vertex (2, negative 2).
    6. A line segment from (negative 3, negative 3) to (0, negative 3) where (0, negative 3) is an open dot. Another line segment exists between (0, 2) and (3, 2). There is an open dot at (3, 2).
  3. Determine the domain and range of the following linear relations.
    1. \(y=x+4\)
    2. \(y=5\)
    3. \(x=7\)
  4. Consider the linear function \(f(x)=-x+2\).
    1. If the domain of this function is \(\mathbb{R}\), determine the range.
    2. If the domain of this function is \(\mathbb{W}\), determine the range.
    3. If the domain of this function is restricted to \(\{x \in \mathbb{R} \mid x \lt -1\}\), determine the range.
    4. If the range of this function is \(\{y \in \mathbb{R} \mid y \le 4\}\), determine the domain.
  5. Determine the domain and range of the following quadratic functions.
    1. \(y=(x-4)^2+5\)
    2. \(y=-(x-1)^2-3\)
    3. \(y=(x-4)(x+2)\)
    4. \(y=-2(x+1)(x-1)\)
    1. Determine the vertex of \(f(x)=x^2+10x+16\) using the indicated method.
      1. Factoring.
      2. Completing the square.
      3. Using the vertex formula.
      4. Partial factoring.
    2. State the range.
  7. Consider the quadratic function \(f(x)=2x^2+3\).
    1. Determine the domain and range.
    2. If the domain is restricted to \(\{x \in \mathbb{R} \mid x \ge -2\}\), determine the range.
    3. If the domain is restricted to \(\{x \in \mathbb{R} \mid x \le -2\}\), determine the range.
  8. In a compressor, the pressure of air, measured in pounds per square inch (psi), is given by the function

    \(P(t)=\dfrac{25}8t^2-25t+100\)

    where \(t\) is the time the compressor is running, in minutes.

    While the compressor is being used, the pressure of air inside the compressor decreases. Once the compressor stops being used, the pressure of air inside the compressor increases until it reaches \(250 \) psi, at which time the compressor stops running. The function \(P(t)=\dfrac{25}8t^2-25t+100\) describes the pressure in the compressor from \(t=0\) until the compressor stops running.

    1. Determine \(P(0)\) and \(P(2)\).
    2. At what time is the pressure the lowest? What is the lowest pressure?
    3. At what time does the pressure reach \(250\) psi?
    4. Determine the domain of \(P(t)\).
    5. Determine the range of \(P(t)\).
    6. Sketch a graph of this function.
  9. Recall that the square root of a negative value is not a real number.
    1. Determine the domain of \(f(x)=\sqrt{x}\).
    2. Determine the domain of \(g(x)=\sqrt{x-5}\).
    3. Determine the domain of \(h(x)=\sqrt{(x-5)(x-1)}\). (Hint: When using set notation, the word 'or' can be used to indicate that either of two conditions can be satisfied for an element to be part of the set).

Lesson 4: Domain and Range of Two New Functions


  1. Determine the domain and range for each function.
    1. \(f(x)=\sqrt{x-5}\)
    2. \(g(x)=\sqrt{1-x}\)
    3. \(h(x)=-3 \sqrt{2x+1}-9\)
  2. The cost, \(C(p)\), in dollars, to manufacture an automobile part is given by \(C(p)=\sqrt{3p-15}+2\), where \(p\) is the retail price in dollars.
    1. For the function \(C(p)\), as \(p\) increases so does \(C(p)\). Give one reason why this is reasonable.
    2. Determine the domain.
    3. Determine the range.
    4. If the retail price is \($8\), what is the cost to manufacture the part?
    5. If the cost to manufacture the part is \($10\), what will be the retail price?
    6. Studies show that the maximum viable retail price for this part is \($53\). State the new domain and range of this function. 
  3. The right angle triangle shown has a hypotenuse of \(10\) cm and a horizontal leg length of \(x\) cm.

    Let \(f(x)\) be the length of the vertical leg.

    1. Determine an expression for \(f(x)\).
    2. Determine the domain and range of \(f(x)\) and interpret these values in the context of the given triangle.
  4. Determine the domain and range of each function.
    1. \(f(x)=\dfrac{4}{3x-9}+1\)
    2. \(g(x)=\dfrac{3}{5-x}-6\)
    3. \(h(x)=\dfrac{-2}{5x-100}-8\)
  5. Determine the domain of each function.
    1. \(f(x)=\dfrac{2}{x^2+x-12}\)
    2. \(g(x)=\dfrac{-5}{x^2+4}+3\)
    3. \(h(x)=\dfrac{-1}{x^2-36}-7\)
  6. Determine the domain and range of \(f(x)=\dfrac{-3}{\sqrt{x+5}}+4\).
  7. Determine the domain of each function.
    1. \(f(x)=\sqrt{x^2-9}\)
    2. \(g(x)=\sqrt{x^2+9}\)
  8. Determine the domain of \(f(x) = \dfrac{1}{\sqrt{x^2+4x+3}}\).