Question

Given the functions f(x)=-4+5 and g(x)=x² + x² - 4x + 5, what type of functions are f(x) and g(x)? Justify your answer. What key feature(s) do f(x) and g(x) have in common?

(Consider domain, range, x-intercepts, and y-intercepts.)

Answer 1

The function f(x) is a constant function with a y-intercept of 5 and a domain and range of all real numbers.

The function g(x) is a quadratic function with a vertex at (1, 7) and a y-intercept of 5. Its domain is all real numbers and its range is all real numbers greater than or equal to 5. It also has x-intercepts at (1, 0) and (3, 0).

The key feature that f(x) and g(x) have in common is the y-intercept of 5. They both intersect the y-axis at the point (0, 5). Additionally, they both have a domain of all real numbers. However, they differ in their range and x-intercepts. The function f(x) has a constant range of {5}, while the function g(x) has a range of all real numbers greater than or equal to 5. The function f(x) does not have any x-intercepts, while the function g(x) has x-intercepts at (1, 0) and (3, 0).

Therefore, f(x) is a constant function and g(x) is a quadratic function. The key feature they have in common is the y-intercept of 5, while they differ in their range and x-intercepts.