Question

Given the functions f(x) = x^3 + x^2 - 2x + 3 and g(x) = log(x) + 2, 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.)

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Answer 1

f(x) is a cubic function and g(x) is a logarithmic function. They have different key features such as domain, range, x-intercepts, and y-intercept.

Step-by-step explanation:

The function f(x) = x^3 + x^2 - 2x + 3 is a polynomial function of degree 3, also known as a cubic function. The function g(x) = log(x) + 2 is a logarithmic function.

Both f(x) and g(x) have different key features:

1. Domain: The domain of f(x) is the set of all real numbers since there are no restrictions on the input values. The domain of g(x) is the set of positive real numbers, as the logarithm function is only defined for positive numbers.
2. Range: The range of f(x) is also the set of all real numbers, as a cubic function does not have any restrictions on the output values. The range of g(x) is the set of all real numbers, as the logarithmic function can output any real number.
3. X-intercepts: To find the x-intercepts of f(x), we set f(x) = 0 and solve for x. To find the x-intercepts of g(x), we set g(x) = 0 and solve for x.
4. Y-intercept: The y-intercept of f(x) can be found by evaluating f(0). The y-intercept of g(x) can be found by evaluating g(0).