Question

module 2
the function (f) is defined by (f(x)=\frac{1}{10}x - 2). what is the (y -)intercept of the graph of (y = f(x)) in the (xy -)plane?
a) ((-2,0))
b) ((0,-2))
c) ((0,\frac{1}{10}))
d) ((\frac{1}{10},0)
for the right - triangle shown, (a = 4) and (b = 5). which expression represents the value of (c)?
a) (4 + 5)
b) (sqrt{(4)(5)})
c) (sqrt{4 + 5})
d) (sqrt{4^{2}+5^{2}}
the function (g) is defined by (g(x)=6x). for what value of (x) is (g(x)=54)?
a producer is creating a video with a length of 70 minutes. the video will consist of segments that are 1 minute long and segments that are 3 minutes long. which equation represents this situation, where (x) represents the number of 1 - minute segments and (y) represents the number of 3 - minute segments?
a) (4xy = 70)
b) (4(x + y)=70)
c) (3x + y = 70)
d) (x + 3y = 70
the function (f) is defined by (f(x)=8x^{2}+4). what is the value of (f(2))?

Explanation:

Step1: Solve the right - triangle problem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), by the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\). Given \(a = 4\) and \(b = 5\), then \(c=\sqrt{4^{2}+5^{2}}\), so the answer to the right - triangle problem is D.

Step2: Find the \(y\) - intercept of the function \(y = f(x)\)

The \(y\) - intercept of a function \(y = f(x)\) is found by setting \(x = 0\). For \(f(x)=\frac{1}{10}x - 2\), when \(x = 0\), \(y=f(0)=\frac{1}{10}(0)-2=-2\). The \(y\) - intercept is the point \((0, - 2)\), so the answer is B.

Step3: Solve for \(x\) in the function \(g(x)\)

Given \(g(x)=6x\) and \(g(x)=54\), we set up the equation \(6x = 54\). Divide both sides by 6: \(x=\frac{54}{6}=9\).

Step4: Find the value of \(f(2)\)

Given \(f(x)=8x^{2}+4\), substitute \(x = 2\) into the function. \(f(2)=8(2)^{2}+4=8\times4 + 4=32 + 4=36\).

Step5: Set up the video - length equation

The length of the video is 70 minutes. The 1 - minute segments contribute \(x\) minutes and the 3 - minute segments contribute \(3y\) minutes. So the equation is \(x + 3y=70\), and the answer is D.

Answer:

1. D
2. B
3. \(x = 9\)
4. \(36\)
5. D