Question

What would be the domain and range of the following g(x) = -|x-3| + 4?

A) Domain: All real numbers, Range: All real numbers
B) Domain: All real numbers, Range: y ≤ 4
C) Domain: x ≤ 3, Range: y ≥ 4
D) Domain: x ≥ 3, Range: y ≤ 4

Answer 1

The domain of the function g(x) = -|x-3| + 4 includes all real numbers, and the range is y ≤ 4, since the inverted 'V' shape of the function due to the negative sign and absolute value dictates that the maximum y-value is 4.

Step-by-step explanation:

The domain of a function is the set of all possible inputs (x-values) the function can accept, and the range is the set of all possible outputs (y-values) the function can produce.

For the function g(x) = -|x-3| + 4, since the absolute value of any real number is defined, the domain includes all real numbers. The negative sign in front of the absolute value flips the graph over the x-axis, and adding 4 shifts the graph up by 4 units. Because the absolute value function creates a 'V' shape, and due to the negative sign, the 'V' shape will be inverted and shifted up. This means the vertex of the 'V' (the highest point) is the maximum of the function, which is at y=4 when x=3.

Therefore, no y-value will ever be greater than 4, making the range of g(x) y ≤ 4. In conclusion, the domain and range of the given function are all real numbers and y ≤ 4, respectively, which corresponds to option B).