Question

which is the graph of the function $f(x) = x^3 + x^2 + 4$?

Explanation:

Step1: Analyze the leading term

The function is \( f(x) = x^3 + x^2 + 4 \). The leading term is \( x^3 \), so as \( x \to +\infty \), \( f(x) \to +\infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \), which matches the end - behavior of the given graph.

Step2: Check the y - intercept

To find the y - intercept, set \( x = 0 \). Then \( f(0)=0^3 + 0^2+4 = 4 \). The graph intersects the y - axis at \( y = 4 \), which is consistent with the given graph.

Step3: Analyze the general shape

The function is a cubic function. The derivative \( f^\prime(x)=3x^2 + 2x=x(3x + 2) \). The critical points are at \( x = 0 \) and \( x=-\frac{2}{3}\). The graph has a local maximum or minimum around these points, and the given graph shows a curve that has a turning point (consistent with the critical points of the cubic function) and the correct end - behavior and y - intercept.

Answer:

The given graph (the one in the image) is the graph of the function \( f(x)=x^3 + x^2+4 \)