Inequalities and Functions

Question

Given the functions f(x) = x³ + x² – 3x + 4 and g(x) = 2^x – 4, 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 first function is a sum of terms with powers of x, thus it is a polynomial function. With a polynomial, any number may be substituted for x and it will still be defined, this the domain is all reals. The second function has the variable x as an exponent, thus it is an exponential function. Likewise any number can be substituted for x with it still being defined, so again the domain is all reals. Thus, the two functions have the same domain, that is something they have in common.

The polynomial is an odd polynomial with a positive coefficient in the leading term, so the end behavior decreases towards negative infinity to the left and increases towards positive infinity towards the right. Thus, the range is all reals.

The exponential has the property that there is an asymptote, because a number raised to a large negative value collapses toward zero. In this case the asymptote is y = -4. As the function goes to the left towards infinity, the function approaches -4. As the function goes to the right towards infinity, it approaches positive infinity. Thus the range for g(x) is the set (-4, positive infinity). The polynomial and the exponential have the same end behavior in the right hand direction.

To find the y intercepts, simply find f(0). (The y intercept occurs at x = 0). f(0) = 4, thus the y intercept of f(x) is 4. g(0) = -3, thus the y intercept of g(x) is -3. (Note that 2 to the power 0 is 1).

The x intercepts occur when y = 0, so to find the x intercepts set the function equal to 0 and solve for y.

Setting f(x) = 0, there does not appear to be a way to factor the polynomial with rational numbers, so there are irrational roots. You can find them by graphing the polynomial on a graphing calculator and selecting the points that touch the x axis.

Setting g(x) = 0, you have 2^x - 4 = 0, 2^x = 4, x = 2. Thus, the x intercept of g(x) is 2.