Question

which of the following is an x - intercept of ( g(x) = 2(x - 10)(x + 100) )?
options:

• ( (1, 0) )
• ( (0, 100) )
• ( (100, 0) )
• ( (10, 0) )
• ( (1000, 0) )
• ( (-10, 0) )

Explanation:

Step1: Recall x-intercept definition

An x-intercept of a function \( g(x) \) is a point where \( y = 0 \), i.e., \( g(x)=0 \). So we set \( g(x) = 0 \) and solve for \( x \).

Step2: Solve \( g(x)=0 \)

Given \( g(x)=2(x - 10)(x + 100) \), set \( 2(x - 10)(x + 100)=0 \). Since 2 ≠ 0, we solve \( (x - 10)(x + 100)=0 \). This gives \( x - 10 = 0 \) or \( x + 100 = 0 \), so \( x = 10 \) or \( x=- 100 \).

Step3: Identify x-intercept points

The x-intercepts are \( (10, 0) \) and \( (-100, 0) \)? Wait, no, wait: Wait, \( x - 10 = 0 \Rightarrow x = 10 \), so point \( (10, 0) \); \( x + 100 = 0 \Rightarrow x=-100 \), so point \( (-100, 0) \)? Wait, but looking at options, there is \( (10, 0) \)? Wait the options: (1,0), (0,100) [y-intercept], (100,0), (10,0), (1000,0), (-10,0). Wait, let's recheck the function: \( g(x)=2(x - 10)(x + 100) \). So set to zero: \( x - 10 = 0 \Rightarrow x = 10 \), so \( (10, 0) \); \( x + 100 = 0 \Rightarrow x=-100 \), but -100 isn't an option. Wait the options: (10,0) is an option (the fourth option: (10,0)). Wait, let's check each option:

• (1,0): Plug x=1 into g(x): 2(1-10)(1+100)=2(-9)(101)≠0.
• (0,100): This is y-intercept (x=0), not x-intercept.
• (100,0): Plug x=100: 2(100-10)(100+100)=2(90)(200)≠0.
• (10,0): Plug x=10: 2(10-10)(10+100)=2(0)(110)=0. So this works.
• (1000,0): 2(1000-10)(1000+100)≠0.
• (-10,0): 2(-10-10)(-10+100)=2(-20)(90)≠0.

So the x-intercept is (10, 0). Wait, but let's check again. Wait the function is \( g(x)=2(x - 10)(x + 100) \). So when x=10, g(10)=2(0)(110)=0, so (10,0) is an x-intercept. When x=-100, g(-100)=2(-110)(0)=0, but -100 isn't an option. So among the options, (10,0) is the x-intercept.

Answer:

(10, 0) (the fourth option: (10, 0))