PCC SLC Math Resources

Problem Set 1

1. \(\begin{aligned}[t] g(\highlight{7})\amp=8-4(\highlight{7})\\ \amp=8-28\\ \amp=-20 \end{aligned}\)
2. \(\begin{aligned}[t] f(\highlight{-3})\amp=-(\highlight{-3})^2+5(\highlight{-3})+12\\ \amp=-9-15+12\\ \amp=-12 \end{aligned}\)
3. \(\begin{aligned}[t] h(\highlight{33})\amp=-\sqrt{\frac{\highlight{33}-25}{2}}\\ \amp=-\sqrt{\frac{8}{2}}\\ \amp=-\sqrt{4}\\ \amp=-2 \end{aligned}\)
4. \(\begin{aligned}[t] t(\highlight{22})\amp=17\\ \end{aligned}\)
5. \(\begin{aligned}[t] y(\highlight{-6})\amp=\abs{-9-(\highlight{-6})}+3\\ \amp=\abs{-3}+3\\ \amp=3+3\\ \amp=6 \end{aligned}\)
6. \(\begin{aligned}[t] z(\highlight{-1})\amp=\frac{\highlight{-1}+6}{(\highlight{-1})^2+1}\\ \amp=\frac{5}{2} \end{aligned}\)
7. \(\begin{aligned}[t] k(\highlight{4})\amp=3\sqrt{21-\highlight{4}^2}\\ \amp=3\sqrt{21-16}\\ \amp=3\sqrt{5} \end{aligned}\)

Problem Set 2

1. Solve \(f(x)=12\) where \(f(x)=4-x\text{.}\)
   \begin{align*} f(x)\amp=12\\ 4-x\amp=12\\ 4-x\subtractright{4}\amp=12\subtractright{4}\\ -x\amp=8\\ \multiplyleft{-1}-x\amp=\multiplyleft{-1}8\\ x\amp=-8 \end{align*}
   The solution set is \(\{-8\}\text{.}\)
2. Solve \(h(t)=100\) where \(h(t)=t^2-21\text{.}\)
   \begin{align*} h(t)\amp=100\\ t^2-21\amp=100\\ t^2-21\addright{21}\amp=100\addright{21}\\ t^2\amp=121\\ t\amp=\pm\sqrt{121}\\ t\amp=\pm 11 \end{align*}
   The solution set is \(\{-11,11\}\text{.}\)
3. Solve \(p(x)=-14\) where \(p(x)=6-\abs{x}\text{.}\)
   \begin{align*} p(x)\amp=-14\\ 6-\abs{x}\amp=-14\\ 6-\abs{x}\subtractright{6}\amp=-14\subtractright{6}\\ -\abs{x}\amp=-20\\ \multiplyleft{-1}-\abs{x}\amp=\multiplyleft{-1}-20\\ \abs{x}\amp=20\\ x\amp=\pm 20 \end{align*}
   The solution set is \(\{-20,20\}\text{.}\)
4. Solve \(w(y)=10\) where \(w(y)=y^2-y-2\text{.}\)
   \begin{align*} w(y)\amp=10\\ y^2-y-2\amp=10\\ y^2-y-2\subtractright{10}\amp=10\subtractright{10}\\ y^2-y-12\amp=0\\ (y-4)(y+3)\amp=0 \end{align*}
   \begin{align*} y-4\amp=0\amp\amp\text{ or }\amp y+3\amp=0\\ y-4\addright{4}\amp=0\addright{4}\amp\amp\text{ or }\amp y+3\subtractright{3}\amp=0\subtractright{3}\\ y\amp=4\amp\amp\text{ or }\amp y\amp=-3 \end{align*}
   The solution set is \(\{-3,4\}\text{.}\)
5. Solve \(r(t)=s(t)\) where \(r(t)=\frac{5}{7}t-3\) and \(s(t)=\frac{2}{3}t+\frac{11}{21}\text{.}\)
   \begin{align*} r(t)\amp=s(t)\\ \frac{5}{7}t-3\amp=\frac{2}{3}t+\frac{11}{21}\\ \multiplyleft{21}(\frac{5}{7}t-3)\amp=\multiplyleft{21}(\frac{2}{3}t+\frac{11}{21})\\ 15t-63\amp=14t+11\\ 15t-63\addright{63}\amp=14t+11\addright{63}\\ 15t\amp=14t+74\\ 15t\subtractright{14t}\amp=14t+74\subtractright{14t}\\ t\amp=74 \end{align*}
   The solution set is \(\{74\}\text{.}\)
6. Solve \(g(x)=y(x)\) where \(q(x)=6-4x^2\) and \(y(x)=(3-x)(8+4x)\text{.}\)
   \begin{align*} g(x)\amp=y(x)\\ 6-4x^2\amp=(3-x)(8+4x)\\ 6-4x^2\amp=24+4x-4x^2\\ 6-4x^2\addright{4x^2}\amp=24+4x-4x^2\addright{4x^2}\\ 6\amp=24+4x\\ 6\subtractright{24}\amp=24+4x\subtractright{24}\\ -18\amp=4x\\ \divideunder{-18}{4}\amp=\divideunder{4x}{4}\\ -\frac{9}{2}\amp=x \end{align*}
   The solution set is \(\{-\frac{9}{2}\}\text{.}\)

Problem Set 3

1. \(f(4)=2\)
2. \(f(2)=2\)
3. \(f(5)\) is not defined.
4. \(f(1)=5\)

1. The solution set is \(\{-1,3\}\text{.}\)
2. The solution set is \(\{-4,-2,0,2,4\}\text{.}\)
3. The solution set is \(\emptyset\text{.}\)
4. The solution set is \(\{-3,1\}\)

Problem Set 4

1. \(g(-1)=6\)
2. \(g(2)=0\)
3. \(g(6)=-8\)
4. \(g(-6)\) is not defined.

1. The solution set is \(\{-2,3\}\text{.}\)
2. The solution set is \(\{\frac{1}{2}\}\text{.}\)
3. The solution set is \(\{-4,5\}\)
4. The solution set is \(\{7\}\text{.}\)

Problem Set 5

1. \(k(2)\) is undefined.
2. \(k(0)-3\)
3. \(k(4)=4\)
4. \(k(-3)=-5\)

1. The solution set is \(\{-1,5\}\text{.}\)
2. The solution set is \(\{4\}\text{.}\)
3. The solution set is \(\{3\}\text{.}\)
4. The solution set is \(\emptyset\text{.}\)

Problem 1

1. To determine the domain of \(y(x)=\sqrt{15-x}\text{,}\) we begin by noting that we cannot take the square root of a negative number (at least over the real numbers). This gives us the following.

   \begin{align*} 15-x \amp\ge 0\\ 15-x\subtractright{15} \amp\ge 0\subtractright{15}\\ -x \amp\ge -15\\ \multiplyleft{-1}-x \amp\le \multiplyleft{-1}-15\\ x \amp\le 15 \end{align*}

   The domain of \(y\) is \((-\infty,15)\)

2. To determine the domain of \(f(t)=\sqrt[3]{t^2-9}\text{,}\) we begin by noting that the polynomial expression \(t^2-9\) is defined for all real numbers as is the cube root function. So the domain of \(f\) is \((-\infty,\infty)\text{.}\)

3. To determine the domain of \(w(x)=\frac{x-7}{x-12}\text{,}\) we begin by noting that we cannot divide by zero. This gives us the following.

   \begin{align*} x-12 \amp\ne 0\\ x-12\addright{12} \amp\ne 0\addright{12}\\ x \amp\ne 12 \end{align*}

   The domain of \(w\) is \((-\infty,12) \cup (12,\infty)\text{.}\)

4. To determine the domain of \(g(x)=\frac{x+3}{x^2+8x+15}\text{,}\) we begin by noting that we cannot divide by zero. This gives us the following.

   \begin{align*} x^2+8x+15 \amp\ne 0\\ (x+3)(x+5) \amp\ne 0 \end{align*}
   \begin{align*} x+3 \amp\ne 0 \amp\amp\text{ and }\amp x+5 \amp\ne 0\\ x+3\subtractright{3} \amp\ne 0\subtractright{3} \amp\amp\text{ and }\amp x+5\subtractright{5} \amp\ne 0\subtractright{5}\\ x \amp\ne -3 \amp\amp\text{ and }\amp x \amp\ne -5 \end{align*}

   The domain of \(g\) is \((-\infty,-5) \cup (-5,-3) \cup (-3,\infty)\text{.}\)

5. To determine the domain of \(r(t)=t^2-3t+9\text{,}\) we begin by noting that \(r\) is a polynomial function and polynomial functions are defined for all real numbers. So the domain of \(r\) is \((-\infty,\infty)\text{.}\)

6. To determine the domain of \(k(t)=\frac{t^2+16}{t^2+16}\text{,}\) we begin by noting that we cannot divide by zero. But \(t^2+16\) is positive for all real number values of \(t\text{,}\) so no value of \(t\) will cause division by zero. So the domain of \(k\) is \((-\infty,\infty)\text{.}\)

Problem 2

The domain is \([-4,5),\text{.}\) The range is \([-1,5]\text{.}\)

Problem 3

The domain is \((-6,\infty)\text{.}\) The range is \((-\infty,6]\text{.}\)

Problem 4

The domain is \((-\infty,0] \cup (2,\infty)\text{.}\) The range is \((-\infty,6]\text{.}\)

1. Simplify \(f(x-7)\) for the function \(f(x)=3x+12\text{.}\)
   \begin{align*} f(\highlight{x-7})\amp=3(\highlight{x-7})+12\\ \amp=3x-21+12\\ \amp=3x-9 \end{align*}
2. Simplify \(g(x)+9\) for the function \(g(x)=14-7x\text{.}\)
   \begin{align*} g(x)\highlightr{+9}\amp=14-7x\highlightr{+9}\\ \amp=23-7x \end{align*}
3. Simplify \(h(5-2t)\) for the function \(h(t)=\sqrt{3-t^2}\text{.}\)
   \begin{align*} h(\highlight{5-2t})\amp=\sqrt{3-(\highlight{5-2t})^2}\\ \amp=\sqrt{3-(5-2t)(5-2t)}\\ \amp=\sqrt{3-(25-20t+4t^2)}\\ \amp=\sqrt{3-25+20t-4t^2}\\ \amp=\sqrt{-4t^2+20t-22} \end{align*}
4. Simplify \(k(x+4)+3\) for the function \(k(x)=2x^2-3\text{.}\)
   \begin{align*} k(\highlight{x+4})\highlightr{+3}\amp=2(\highlight{x+4})^2-3\highlightr{+3}\\ \amp=2(x+4)(x+4)\\ \amp=2(x^2+8x+16)\\ \amp=2x^2+16x+32 \end{align*}
5. Simplify \(r(\sqrt{t-4})\) for the function \(r(t)=3-7t^2\text{.}\)
   \begin{align*} r(\highlight{\sqrt{t-4}})\amp=3-7(\highlight{\sqrt{t-4}})^2\\ \amp=3-7(t-4)\\ \amp=3-7t+28\\ \amp=31-7t \end{align*}
6. Simplify \(y(t+5)-y(t-3)\) for the function \(y(t)=3t-16\text{.}\)
   \begin{align*} y(\highlight{t+5})-y(\highlightb{t-3})\amp=(3(\highlight{t+5})-16)-(3(\highlightb{t-3})-16)\\ \amp=3t+15-16-(3t-9-16)\\ \amp=3t-1-(3t-25)\\ \amp=3t-1-3t+25\\ \amp=24 \end{align*}
7. Simplify \(w(x-2)+w(2-x)+2\) for the function \(w(x)=5-x\text{.}\)
   \begin{align*} w(\highlight{x-2})+w(\highlightb{2-x})\highlightr{+2}\amp=(5-(\highlight{x-2}))+(5-(\highlightb{2-x}))\highlightr{+2}\\ \amp=5-x+2+5-2+x+2\\ \amp=12 \end{align*}
8. Simplify \(s(t-8)\) for the function \(s(t)=\frac{t+8}{t}\text{.}\)
   \begin{align*} s(\highlight{t-8})\amp=\frac{\highlight{t-8}+8}{\highlight{t-8}}\\ \amp=\frac{t}{t-8} \end{align*}
9. Simplify \(3u(2x)\) for the function \(u(x)=-x^2\text{.}\)
   \begin{align*} \highlightg{3}u(\highlight{2x})\amp=\highlightg{3\cdot}-(\highlight{2x})^2\\ \amp=-3 \cdot 4x^2\\ \amp=-12x^2 \end{align*}
10. Simplify \(\frac{1}{2}h(2t-7)-8\) for the function \(h(t)=8t-12\text{.}\)
    \begin{align*} \highlightg{\frac{1}{2}}h(\highlight{2t-7})\highlightr{-8}\amp=\highlightg{\frac{1}{2}}(8(\highlight{2t-7})-12)\highlightr{-8}\\ \amp=\frac{1}{2}(16t-56)-20\\ \amp=8t-28-20\\ \amp=8t-48 \end{align*}

Problem Set 1

1. The solution set is \(\{x \mid -2 \lt x \lt 0 \text{ or } 2 \lt x \lt 4\}\text{.}\)

   The solution set is \((-2,0) \cup (2,4)\text{.}\)

2. The solution set is \(\{x \mid -4 \leq x \leq -2 \text{ or } 0 \leq x \leq 2 \text{ or } 4 \leq x \lt 5\}\text{.}\)

   The solution set is \([-4,-2] \cup [0,2] \cup [4,6)\text{.}\)

3. The solution set is \(\{\}\text{.}\)

   The solution set is \(\emptyset\text{.}\)

4. The solution set is \(\{x \mid -4 \leq x \lt 5\}\text{.}\)

   The solution set is \([-4,5)\text{.}\)

Problem Set 2

1. The solution set is \(\{-2 \leq x \leq 3\}\text{.}\)

   The solution set is \([-2,3]\text{.}\)

2. The solution set is \(\{x \mid -6 \lt x \lt 2 \text{ or } x \geq \frac{1}{2}\}\text{.}\)

   The solution set is \((-6,-2) \cup [\frac{1}{2},\infty)\text{.}\)

3. The solution set is \(x \mid x \gt 5\}\text{.}\)

   The solution set is \((5,\infty)\text{.}\)

4. The solution set is \(\{x \mid -6 \lt x \lt -4 \text{ or } -4 \lt x \lt 5\}\text{.}\)

   The solution set is \((-6,-4) \cup (-4,5)\text{.}\)

Problem Set 3

1. The solution set is \(\{x \mid -1 \leq x \lt 0 \text{ or } 2 \lt x \leq 5\}\text{.}\)

   The solution set is \([-1,0) \cup (2,5]\text{.}\)

2. The solution set is \(\{x \mid x \leq 0 \text{ or } x \geq 4\}\text{.}\)

   The solution set is \((-\infty,0] \cup [4,\infty)\text{.}\)

3. The solution set is \(\{\}\text{.}\)

   The solution set is \(\emptyset\text{.}\)

4. The solution set is \(\{x \mid x \leq 0 \text{ or } x \gt 2\}\text{.}\)

   The solution set is \((-\infty,0] \cup (2,\infty)\text{.}\)