Question

determine whether each statement is always, sometimes, or never true. an absolute value function of the form f(x)=/x+a/+b has exactly one x intercept.

Answer 1

The statement that f(x) = |x+a| + b has exactly one x-intercept is sometimes correct.

Solution-

It solely depends on b, whether the function will have one or two or zero x intercept.

This plot of the given function, f(x) = |x+a| + b will be the basic absolute value graph i.e V shape, with vertex translated to (-a, b), instead of origin.

1- If b is zero, the graph will have exactly one x-intercept, at x= -a

2- If b is positive, the whole graph will be above the x-axis, hence it will have no x-intercepts.

3- If b is negative, the graph will be below the x-axis, hence it will have two x-intercepts.

∴ The statement is sometimes true.