Math 2 Exam I Practice

These are a collection of problems you can use to practice for the exam. To see the answers go to this page.

P-level

1. Simplify the following sets. Write your simplified answer using interval notation.
   1. \((-\infty,4)\cap [-2,9]\)
   2. \([2,6)\cup [1,5]\)
   3. \((1,4)\setminus [2,3]\)
2. Determine the domain and range of the following function \(f\) whose graph is given below.

3. Identify the slope of the lines given below.

   1. \(y=-\frac{1}{3}x+4\)
   2. \(y-3=2(x+4)\)

4. Take \(f\) and \(g\) to be functions with the given domains. Identify the domain of \(f-g\) and \(fg\). Write your answer as a union of intervals.

   1. \(\mathcal{D}(f)=(-1,4)\) and \(\mathcal{D}(g)=(-\infty,6]\)
   2. \(\mathcal{D}(f)=[5,\infty)\) and \(\mathcal{D}(g)=[-4,6)\)

5. Identify the domain of the following functions. Write your answer as a union of intervals.

   1. \(f(x)=\frac{5}{2}x\)
   2. \(f(x)=x^3+5x+1\)
   3. \(f(x)=\sqrt{-x+4}\)
   4. \(f(x)=\dfrac{x}{x+3}\)

6. Identify the domain of the function \(f\) given by \[f(x)=\begin{cases}-2x+1&\text{if }x<-2\\ x&\text{if }4<x<5.\end{cases}\]

7. Compute the sum \(\langle 2,-4\rangle+(3,2).\)

8. Determine the vector that moves the point \((1,2)\) to the point \((3,1).\)

9. Compute the following: \(2\langle 3, 1\rangle.\)

10. Determine the center and radius of the circle whose equation is given by \[(x-2)^2+(y+9)^2=16.\]

11. A line \(L\) has slope \(\frac{4}{3}\). Identify a vector that moves points along \(L\).

12. Take \(V=\langle 2,3\rangle.\) Calculate \(V_{\perp}.\)

13. A line \(L\) has slope \(\frac{4}{3}\). Identify the slope of the line perpendicular to \(L\).

14. Take \(L\) to be the line with slope \(3\). Determine the slope of \(L^{-1}\).

15. Take \(f\) to be an invertible function with a domain of \((-2,9]\) and a range of \((2,\infty)\). Determine the domain and range of \(f^{-1}\).

C-level

1. Simplify the following sets. Write your simplified answer using interval notation.

   1. \(\left((-\infty,4)\cap [-2,9]\right)\cup(-1,5]\)
   2. \(\left([2,6)\cup [1,5]\right)\cap(-\infty,1)\)
   3. \(\left((1,4)\setminus [2,3]\right)\cup\left(\frac{3}{2},5\right]\)

2. Identify the \(y\)-intercept and \(x\)-intercept of the lines given below.

   1. \(y=-\frac{1}{3}x+4\)
   2. \(y-3=2(x+4)\)
   3. the line that passes through \((1,0)\) and \((2,5)\)

3. Take \(f\) and \(g\) to be functions with the given domains and zero sets \(Z(f)\) and \(Z(g)\). Identify the domain of \(\frac{f}{g}\) and \(\frac{g}{f}\). Write your answer as a union of intervals.

   1. \(\mathcal{D}(f)=(-1,4)\), \(\mathcal{D}(g)=(-\infty,6]\), \(Z(f)=\left\{\frac{1}{2},3\right\}\), and \(Z(g)=\left\{-1,5\right\}\)
   2. \(\mathcal{D}(f)=[5,\infty)\), \(\mathcal{D}(g)=[-4,6)\), \(Z(f)=\left\{5,11\right\}\), and \(Z(g)=\left\{\frac{11}{2},\frac{13}{2}\right\}.\)

4. Take \(a\), \(b\), \(c\) and \(d\) to be the functions given by \[a(x)=x,\quad b(x)=x^2,\quad c(x)=x-1,\quad d(x)=3x,\text{and}\quad e(x)=\sqrt{x}.\] Decompose the function \(f\) into sums, products, quotients and or composites of \(a,b,c,d,\) and \(e\), where \(f\) is given by \[f(x)=\frac{x^2-1}{x}+\sqrt{x^2+3x}\]

5. Sketch the function \(f\) given by \[f(x)=\begin{cases}-2x+1&\text{if }x<-2\\ x&\text{if }4<x<5.\end{cases}\]

6. Determine the projection of the point \((-3,5)\) onto the unit circle.

7. Determine the point on the line \(L\) given by \(y=3x+4\) that is closest to the point \((1,2).\)

8. Take \(f\) to be the function given by \(f(x)=(x+2)^3+1\). Find the inverse of \(f\) and its domain and range.

B-level

1. Simplify the following sets. Write your simplified answer using interval notation.

   1. \[\biggr([1,3)\cup(2,\infty)\biggr)\cap\biggr( (0,4]\cup(-\infty,2)\biggr)\]
   2. \[\biggr((-\infty,3)\cup(2,\infty)\biggr)\cap(-4,4]\biggr)\cup\biggr( (-3,8]\cap(-\infty,2]\biggr)\]

2. A particle moves at a constant velocity on the time intervals \([0, 10]\) and \((10, 12].\) It is at position \((2, -1)\) at time \(0\), at position \((4, 10)\) at time \(10\), and at position \((11, 0)\) at time \(12\). Determine an equation for the position, \(\ell(t)\), of the particle at time \(t.\)

3. The line \(L\) contains the points \((3,11)\) and \((2,16)\). Identify all points on \(L\) that are a distance of \(3\) from \((3,11)\).

4. Determine the projection of the point \((5,4)\) onto \(\mathcal{C}_3(2,0)\).

5. Take \(f\) and \(g\) to be the functions given \[f(x)=\begin{cases}x &\text{ if } x<3\\3x+1 &\text{ if } x\geq 3\end{cases}\quad\text{and}\quad g(x)=\begin{cases}-2x &\text{ if } x<1\\ x+10 &\text{ if } x\geq 1\end{cases}.\] Solve \(f(x)>g(x)\) without using graphing software. Write your answer using interval notation.

A-level

1. Take \(f\) to be the function given by \[f(x)=\begin{cases}3x &\text{ if } x<2\\x^2-1 &\text{ if } 5\leq x< 7\end{cases}\] and \(g\) to be the piecewise linear function whose graph is given below. Write \(f\circ g\) as a piecewise function and state the domain of \(f\circ g\).

2. Take \(f\) to be the function whose graph is given below.
   1. Determine where \(f(x)<0\). Write your answer as a union of intervals.
   2. Determine where \(f(x)\geq 1.\) Write your answer as a union of intervals.
   3. Determine where \(-1\leq f(x)<1\). Write your answer as a union of intervals.

3. Take \(f\) to be the function given by \(f(x)=x^2\) defined on \(\mathcal{D}(f)=(-4,-3]\cup(-2,-1)\cup [0,1].\)
   1. Sketch \(f\) and \(f^{-1}\).
   2. Write a formula for \(f^{-1}\).
4. Take \(L\) to be the line that is given by \(y=3x+1.\) Determine the reflection \(L\) across the line \(M\) that is given by \(y=2x-1.\)

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