MHF4U Combining Functions Practice Test (with Solutions)

Free Combining Functions Practice Test

8 questions (24 parts), exactly as a free account gets them in the first Combining Functions test on Martian Lab. Try each one on paper first, then open the answer.

1. Given $f(x) = (x-1)^2 + 2$ and $g(x) = -x^2 + 2$. Find the equation of the following combine functions. Simplify if appropriate.

   1. $f(x) + g(x)$
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      $= x^2 - 2x + 1 + 2 - x^2 + 2$ $= -2x + 5$
   2. $g(x) - f(x)$
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      $= -x^2 + 2 - x^2 + 2x - 1 - 2$ $= -2x^2 + 2x - 1$
   3. $f(x)g(x)$
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      $= [(x-1)^2+2]\cdot(-x^2+2)$
   4. $\dfrac{f(x)}{g(x)}$
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      $= \dfrac{x^2 - 2x + 3}{-x^2 + 2}$
   5. $f \circ g$
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      $= f(g(x))$ $= [(-x^2+2)-1]^2 + 2$
   6. $g(f(x))$
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      $= f \circ g$ $= -[(x-1)^2+2]^2 + 2$
2. Let $f = \{(-4,-4),(2,4),(1,0),(3,5),(4,6)\}$ and $g = \{(-4,2),(0,2),(1,2),(2,2),(4,4)\}$. Determine each of the following combine functions as a set of ordered pairs if possible.

   1. $f + g$
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      $= \{(-4,-2),(2,6),(1,2),(4,10)\}$
   2. $f - g$
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      $\{(-4,-6),(2,2),(1,-2),(4,2)\}$
   3. $f \times g$
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      $= \{(-4,-8),(2,8),(1,0),(4,24)\}$
   4. $g \div f$
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      $= \left\{\left(-4,\tfrac{1}{2}\right),\left(2,\tfrac{1}{2}\right),\left(4,\tfrac{2}{5}\right)\right\}$
   5. $f(g(4))$
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      $= f(4)$ $= 6$
   6. $g(f^{-1}(4))$
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      $= g(2)$ $= 2$
3. Let $f(x) = x - 1$ and $g(x) = -2x^2 - 5$, find $x$ such that $f \circ g = g \circ f$.

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   $f \circ g = f(g(x)) = f(-2x^2-5) = (-2x^2-5)-1 = -2x^2-6$ $g \circ f = g(f(x)) = g(x-1) = -2(x-1)^2 - 5 = -2x^2 + 4x - 7$ $\therefore\ -2x^2 - 6 = -2x^2 + 4x - 7 \; , \; 4x = 1, \; x = \dfrac{1}{4}$
4. Let $f(x) = \log_2 x$ and $g(x) = x + 4$. Sketch $h(x) = (f \circ g)(x)$ and determine its domain and range.

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   $f \circ g(x) = f(x+4) = \log_2(x+4)$ $D: \{x \in \mathbb{R},\ x > -4\}$ $R: \{y \in \mathbb{R}\}$
5. A company produces a toy for $9.45 per unit, plus a fixed operating cost of $52,000. The company sells the toy for $15.80 per unit.

   1. Determine a function, $C(x)$, to represent the cost of producing $x$ units.
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      $C(x) = 9.45x + 52000$
   2. Determine a function, $I(x)$, to represent income from sales of $x$ units.
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      $I(x) = 15.80x$
   3. Determine a function that represents the profit, $P(x)$.
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      $P(x) = I(x) - C(x) = 6.35x - 52000$
   4. How many units must be sold for the company to break even?
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      $P(x) = 0$ $6.35x - 52000 = 0$ $x = 8188.97$ $\therefore\ 8189$ units must be sold.
6. For the functions $f(x)$ and $g(x)$ are given in the graph. Find the indicated corresponding function values.

   

   1. $(f+g)(1)$
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      $(f+g)(1) = 0.5 + 4 = 4.5$
   2. $(f-g)(2)$
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      $(f-g)(2) = 1 - 4 = -3$
   3. $(f \div g)(0)$
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      $(f \div g)(0) =$ undefined
   4. $(f \times g)(2)$
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      $(f \times g)(2) = 1 \times 4 = 4$
7. A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of 9 cm per second. Express the surface area of the balloon as a function of time $t$ (in seconds) and simplify if it is appropriate. Note: Surface area of Sphere $= 4\pi r^2$

   Show answer

   $r = 9t$ $A = 4\pi r^2 = 4\pi(9t)^2 = 324\pi t^2 \; (\text{cm}^2)$ $\therefore$ The surface area of the balloon is $324\pi t^2\text{ cm}^2$.
8. Let $f(x) = mx^2 + 2x + 5$ and $g(x) = 2x^2 - nx - 2$. The functions are combined to form the new function $h(x) = f(x) \div g(x)$. Points $(1,6)$ and $(-1,10)$ satisfy the new function. Determine the values of $m$ and $n$.

   Show solution

   $h(x) = \frac{mx^2 + 2x + 5}{2x^2 - nx - 2}$ $h(1) = 6 \; \dfrac{m+7}{-n} = 6 \; , \; m+7 = -6n$ $h(-1) = 10 \; \dfrac{m+3}{n} = 10 \; , \; m+3 = 10n$ $4 = -16n,\ n = \dfrac{-1}{4}$ $m = \dfrac{-11}{2}$

This Unit on Martian Lab

The lessons cover the four ways to combine functions, then composition.

1. Combining Functions
2. Composition of Functions

Then 4 full Combining Functions practice tests to check yourself.

Combining Functions FAQ

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