1.
Item 1. Which of the following graph is that of a function and its inverse?
Solution:A function and its inverse are reflections of each other across the line \(y=x\).The matching pair must appear the same distance from the line \(y=x\), but on opposite sides.Among the choices, Graph IV shows this inverse relationship.Answer: D
2.
Item 3. The graph of \(y=f(x)\) is shown below. If the function is changed to \(y=f\left(\frac{2}{5}x\right)\), what will be point B's new coordinates?
Solution:Replacing \(x\) with \(\frac{2}{5}x\) causes a horizontal expansion by a factor of \(\frac{5}{2}\).This multiplies every \(x\)-value by \(\frac{5}{2}\), while the \(y\)-value stays the same.Point B is \((0,1)\).\[0\cdot\frac{5}{2}=0\]Therefore, point B stays at \((0,1)\). Since point B is on the \(y\)-axis, it is an invariant point.Answer: E
3.
Item 2. Which of the following is the inverse equation of \(y=3x+1\)?
Solution:Start with the original equation:\[y=3x+1\]Interchange \(x\) and \(y\):\[x=3y+1\]Solve for \(y\):\[x-1=3y\]\[y=\frac{x-1}{3}\]Therefore, the inverse equation is \(y=\frac{x-1}{3}\).Answer: D
4.
Item 5. What happens to the graph of a function if you replace \(x\) with \(5x\) and \(y\) with \(5y\)?
Solution:Replacing \(x\) with \(5x\) causes a horizontal compression by a factor of \(\frac{1}{5}\).Replacing \(y\) with \(5y\) causes a vertical compression by a factor of \(\frac{1}{5}\).Therefore, both horizontal and vertical directions are compressed by \(\frac{1}{5}\).Answer: D
5.
Item 6. Prepare for transformation the following equation: \(4y=8|-3x+6|-28\).
Solution:Start with:\[4y=8|-3x+6|-28\]Divide both sides by \(4\):\[y=2|-3x+6|-7\]Factor the coefficient of \(x\) inside the absolute value:\[-3x+6=-3(x-2)\]Substitute back:\[y=2|-3(x-2)|-7\]Answer: A
6.
Item 4. The graph of \(y=f(x)\) is shown. If the function is changed to \(y=f(2x)\), then the transformed graph is best illustrated by:
Solution:Replacing \(x\) with \(2x\) causes a horizontal compression by a factor of \(\frac{1}{2}\).This means each \(x\)-coordinate is multiplied by \(\frac{1}{2}\), while the \(y\)-coordinate stays the same.For example:\[(-2,3)\rightarrow(-1,3)\]The graph that shows this horizontal compression is Graph D.Answer: D
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