🔬 Tutorial problems gamma \(\gamma\)#
Note
This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.
\(\gamma\).1#
Suppose that the cost of producing \(Q\) units of a commodity is given by \(C(Q) = 1, 000 + 300Q + Q^2\).
Compute \(C(0)\), \(C(100)\), and \(C(101) - C(100)\).
Compute \(C(Q + 1) - C(Q)\) and explain the meaning of this expression.
[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 4.2, Question 6
\(\gamma\).2#
Consider a rule of association that assigns to each person their sibling. Is this rule a function? Why or why not?
To properly answer the question it is important start with the definition of domain and co-domain of the described rule of association. Properly stated problem would have defined both explicitly, but if not, the answer should include several possible cases.
\(\gamma\).3#
Consider the function \(f(x) = \frac{3x+6}{x-2}\).
Find the domain of this function
Show that \(5\) belongs to the range of this function.
Show that \(3\) does not belong to the range of this function.
[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 4.2, Question 14
To show that a particular function value is in range, find a value for \(x\) corresponding to it and observe it is in the domain of the function.
\(\gamma\).4#
If \(f(x)=3x-x^3\) and \(g(x)=x^3\), compute the following:
\((f+g)(x)\)
\((f-g)(x)\)
\((fg)(x)\)
\((f/g)(x)\)
\(f(g(1))\)
\(g(f(1))\)
\((f \circ g)(x)\)
\((g \circ f)(x)\)
[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 5.2, Question 3
For standard arithmetic operations \(\bullet\) notation \((f \bullet g)(x)\) is equivalent to \(f(x) \bullet g(x)\) point by point for all \(x\) in the intersection of the domains for \(f(x)\) and \(g(x)\).
\(\gamma\).5#
Let \(f(x)=3x+7\). Compute \(f(f(x))\), and find the value \(x^\star\) such that \(f(f(x^\star))=100\).
[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 5.2, Question 3
\(\gamma\).6#
Find the domains of the following functions and then illustrate those domains in the \((x, y)\)–coordinate-plane.
\(f(x,y)= \frac{x^2+y^3}{y-x+2}\)
\(f(x,y)=\sqrt{2-(x^2 +y^2)}\)
\(f(x,y)=\sqrt{(4-x^2 -y^2)(x^2 +y^2 -1)}\)
[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 11.1, Question 6
Disregard the fact that the questions asks about function of two variables and focus on finding the domain of these functions in the form of a restriction on \((x,y)\).