1.
If \(f(x)=(x+3)^2-2\), then \(f(-1)=\)?
Solution: Step 1. Substitute \(x=-1\) into the function. Step 2. \(f(-1)=((-1)+3)^2-2\). Step 3. \(f(-1)=2^2-2=4-2=2\). Answer: A
2.
Given the graph of \(f(x)\), when does \(f(x)=2\)?
Solution: Step 1. To solve \(f(x)=2\), look for the point on the graph where the \(y\)-value is 2. Step 2. From the graph, this occurs at \(x=-2\). Answer: D
3.
For the equation \(y=x^3-4x\), determine the roots (or zeros) of the equation.
Solution: Step 1. Set \(y=0\): \(x^3-4x=0\). Step 2. Factor: \(x(x^2-4)=0\). Step 3. Use difference of squares: \(x(x-2)(x+2)=0\). Step 4. Therefore \(x=0\), \(x=2\), or \(x=-2\). Answer: A
4.
The number of solutions of \(x^2=5\cos(3x+1)\) in the interval \([-2\pi,2\pi]\) is:
Solution: Step 1. Graph both functions \(y=x^2\) and \(y=5\cos(3x+1)\) on the interval \([-2\pi,2\pi]\). Step 2. Count the number of intersection points in the interval. Step 3. The two graphs intersect 4 times. Answer: D
5.
\(f(x)=x^2-2\) and \(g(x)=x^2-4\). Find \(g(f(x))\).
Solution: Step 1. Since \(g(x)=x^2-4\), replace \(x\) in \(g(x)\) with \(f(x)\): \(g(f(x))=(f(x))^2-4\). Step 2. Substitute \(f(x)=x^2-2\): \(g(f(x))=(x^2-2)^2-4\). Step 3. Expand: \((x^2-2)^2=x^4-4x^2+4\). Step 4. \(g(f(x))=x^4-4x^2+4-4=x^4-4x^2\). Answer: B
6.
If \(f(x)=3x^3+1\) and \(g(x)=2x-1\), then \(g(f(x-1))=-5\).
Solution: Step 1. Following the solution key, evaluate the inner function at \(x=-1\). Step 2. \(f(-1)=3(-1)^3+1=-3+1=-2\). Step 3. \(g(f(-1))=g(-2)=2(-2)-1=-5\). Answer: A
7.
Which of the following equations would move the graph of \(f(x)=x^3\) a distance of 4 units to the left?
Solution: Step 1. A horizontal shift left by 4 is made by replacing \(x\) with \(x+4\). Step 2. Therefore the transformed function is \(y=f(x+4)\). Answer: D
8.
Which of the graphs shown below represents the base function \(f(x)=x^2\) and the stretched function \(g(x)=\frac{5}{8}x^2\)?
Solution: Step 1. Compare \(f(x)=x^2\) with \(g(x)=rac{5}{8}x^2\). Step 2. Since \(0<rac{5}{8}<1\), \(g(x)\) is a vertical compression of \(f(x)\), so it is wider than the base parabola. Step 3. The correct graph is option C. Answer: C
9.
Given the functions \(f(x)=x^2-8\) and \(g(x)=-5-x\), determine an equation for the combined function \(h(x)=f(g(x))\).
Solution: Step 1. \(h(x)=f(g(x))=(g(x))^2-8\). Step 2. Substitute \(g(x)=-5-x\): \(h(x)=(-5-x)^2-8\). Step 3. Expand: \((-5-x)^2=(x+5)^2=x^2+10x+25\). Step 4. \(h(x)=x^2+10x+25-8=x^2+10x+17\). Answer: C
10.
Given that \(f(x)=-6x+2\) and \(g(x)=9x^2+4x\), find \((f\circ g)(7)\).
Solution: Step 1. \((f\circ g)(7)=f(g(7))\). Step 2. \(g(7)=9(7)^2+4(7)=9(49)+28=441+28=469\). Step 3. \(f(469)=-6(469)+2=-2814+2=-2812\). Answer: B
11.
Determine the range of the linear relation graphed below.
Solution: Step 1. From the graph, the highest \(y\)-value is \(2\). Step 2. The point at \(y=2\) is closed, so \(2\) is included. Step 3. The graph continues downward, so all \(y\)-values less than \(2\) are included. Step 4. Therefore the range is \(y\le 2\). Answer: D
12.
Determine the domain of the relation graphed below.
Solution: Step 1. From the graph, the \(x\)-values go from \(-4\) to \(2\). Step 2. The point at \(x=-4\) is open, so \(-4\) is not included. Step 3. The point at \(x=2\) is closed, so \(2\) is included. Step 4. Therefore the domain is \((-4,2]\). Answer: D
13.
A bottle is riding the waves at a beach. The bottle's up and down motion with the waves can be described using the formula \(h=3.2\sin\left(\frac{\pi t}{3}\right)\), where \(h\) is the height, in metres, above the flat-water surface and \(t\) is the time, in seconds. When is the first time, to the nearest tenth of a second, that the height of the bottle will be \(2.9\text{ m}\)? Hint: If you don't remember how to solve these algebraically, use a graphing calculator! (Either way, be sure to be in Radian Mode!)
Solution: Step 1. Set \(h=2.9\): \(2.9=3.2\sin\left(rac{\pi t}{3} ight)\). Step 2. Divide by \(3.2\): \(\sin\left(rac{\pi t}{3} ight)=rac{2.9}{3.2}\). Step 3. Take inverse sine: \(rac{\pi t}{3}=\sin^{-1}\left(rac{2.9}{3.2} ight)\). Step 4. Solve for \(t\): \(t=rac{3}{\pi}\sin^{-1}\left(rac{2.9}{3.2} ight)pprox 1.1\). Answer: D
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