Question

​The zeros of a rational function g are -4 and 3. which of the following expressions could define g(x)?

1) g(x) = (x + 4) / (x - 3)
2) g(x) = (x - 4) / (x + 3)
3) g(x) = (x - 4) / (x - 3)
4) g(x) = (x + 4) / (x + 3)

Answer 1

To find the expression that defines the rational function g(x) with zeros -4 and 3, we substitute these values into the given options. The only option that results in zero for both values is option 4.

Step-by-step explanation:

The zeros of a rational function are the values of x that make the function equal to zero. In this case, the zeros are -4 and 3. To determine which expression defines the function g(x), we need to check if plugging in these values into the expression results in zero.

1. For option 1, g(x) = (x + 4) / (x - 3), if we substitute -4, we get (-4 + 4) / (-4 - 3) = 0 / -7 = 0, and if we substitute 3, we get (3 + 4) / (3 - 3) = 7 / 0, which is undefined. So option 1 does not define g(x).
2. For option 2, g(x) = (x - 4) / (x + 3), if we substitute -4, we get (-4 - 4) / (-4 + 3) = -8 / -1 = 8, and if we substitute 3, we get (3 - 4) / (3 + 3) = -1 / 6. So option 2 does not define g(x).
3. For option 3, g(x) = (x - 4) / (x - 3), if we substitute -4, we get (-4 - 4) / (-4 - 3) = -8 / -7 = 8/7, and if we substitute 3, we get (3 - 4) / (3 - 3) = -1 / 0, which is undefined. So option 3 does not define g(x).
4. For option 4, g(x) = (x + 4) / (x + 3), if we substitute -4, we get (-4 + 4) / (-4 + 3) = 0 / -1 = 0, and if we substitute 3, we get (3 + 4) / (3 + 3) = 7 / 6. So option 4 does define g(x).

Therefore, the correct expression that defines g(x) is

g(x) = (x + 4) / (x + 3)

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