Question

A function g(x) has x-intercepts at (StartFraction 1 Over 2 EndFraction, 0) and (6, 0). Which could be g(x)? g(x) = 2(x 1)(x 6) g(x) = (x – 6)(2x – 1) g(x) = 2(x – 2)(x – 6) g(x) = (x 6)(x 2).

Answer 1

You can use the fact that if graph of function of x has intercept at x axis at point x = a, then putting a in that function will make function 0.

From the given options, g(x) could be given by:

Option B: g(x) = (x-6)(2x-1)

How to know if given function has x intercept at x = a?

If you input that function value a, and that function outputs the value 0, then the plot will be at point (a,0) which is on x axis. Thus graph will cut x axis at this point, thus, this point will be called x intercept by graph of given function.

Thus, to check if a function has x intercept at x = a, put this x=a value as input and see if it evaluates to 0 or not. If it does evaluate to 0, then yes it has x intercept at x = a, else not.

Checking all options to see if they have x intercept at (1/2, 0)

$g(x) = 2(x+1)(x+6)$

At x = 1/2, we get:

$g(\dfrac{1}{2}) = 2(\dfrac{1}{2} + 1)(\dfrac{1}{2} + 6) = 3 \times \dfrac{13}{2} = \dfrac{39}{2} \neq 0$

$g(x) = (x-6)(2x-1)$

At x = 1/2, we get:

$g(\dfrac{1}{2}) = (\dfrac{1}{2} - 6)(2 \times \dfrac{1}{2} - 1) = (\dfrac{1}{2} - 6})(1 - 1) = \dfrac{1}{2} - 6) \times 0 = 0$

Thus, this can be the function g(x).

$g(x) = (x-6)(2x-1)$

At x = 1/2, we get:

$g(1/2) =(1/2 - 2)(1/2 - 6) = -3/2 \times -11/2 = 33/4 \neq 0$

$g(x) = (x+6)(x+2)$

At x = 1/2, we get:

$g(1/2) = (1/2 + 6)(1/2 + 2) > 0$

Thus, only second option g(x) = (x-6)(2x-1) can be the given function with x intercept at (1/2, 0)

One other trick which can serve as a short cut, can be that since the g(x) options are written all factored, then one factor in correct option will be such that putting x = 1/2 will result it to be 0.

x=  1/2 -> 2x  = 1 or (2x - 1) = 0. Thus, one factor in g(x) if is (2x-1), then that has x intercept at x = 1/2. Sometimes you may need to simply and factorize g(x) to see such stuffs.

Thus, only second option g(x) = (x-6)(2x-1) can be the given function with x intercept at (1/2, 0)

Learn more about x-intercept here:

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