The range of f(x) = |x| is y ≥ 0. If a < 0 and b ≠ 0 for g(x) = a|x| + b, what is the range of function g?

The required range of the function g(x) = a|x| + b is y ≥ 0.

Let's analyze the function g(x) = a|x| + b step by step to determine its range.

The function |x| represents the absolute value of x, which means it always yields non-negative values. Therefore, a|x| will also be non-negative for any value of x.

Adding b to a non-negative value will shift the entire range up by b units, but it will still be non-negative.

Since a < 0, a|x| will be negative when x is positive. However, the absolute value function ensures that it will be positive when x is negative, balancing out the effect.

Finally, adding the constant term b to a|x| makes the entire range of g(x) non-negative.

In summary, the range of g(x) = a|x| + b is y ≥ 0.

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