Question

question 9 (1 point)
given the functions $f(x) = x^3 - 1$ and $g(x) = x + 2$, a graph of the combined function $y = f(g(x))$ most likely resembles.

Explanation:

Step1: Find the composition \( f(g(x)) \)

Substitute \( g(x) \) into \( f(x) \). Since \( g(x)=x + 2 \) and \( f(x)=x^{3}-1 \), we have:
\( f(g(x))=f(x + 2)=(x + 2)^{3}-1 \)

Step2: Expand \( (x + 2)^{3} \)

Using the formula \( (a + b)^{3}=a^{3}+3a^{2}b + 3ab^{2}+b^{3} \) with \( a = x \) and \( b = 2 \):
\( (x + 2)^{3}=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}=x^{3}+6x^{2}+12x + 8 \)

Step3: Simplify \( f(g(x)) \)

Subtract 1 from the expanded form:
\( f(g(x))=x^{3}+6x^{2}+12x + 8-1=x^{3}+6x^{2}+12x + 7 \)
This is a cubic function. The leading term is \( x^{3} \), so as \( x
ightarrow\infty \), \( y
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \). The graph will have a single inflection point (since it's a cubic) and the shape will be similar to the graph of a cubic function with a positive leading coefficient, shifted and stretched due to the coefficients of the quadratic and linear terms. To identify the graph, we can also analyze key points or the general shape of cubic functions.

(Note: Since the actual graphs are not provided here, the process above shows how to determine the form of \( f(g(x)) \) to match with the given graph options. The key is recognizing the composition of functions, expanding it, and then analyzing the resulting cubic function's properties like end - behavior, degree, and leading coefficient to identify the correct graph.)

Answer:

To determine the graph of \( y = f(g(x))=(x + 2)^{3}-1=x^{3}+6x^{2}+12x + 7 \), we analyze its properties: it is a cubic function with leading coefficient \( 1>0 \), so as \( x
ightarrow\infty \), \( y
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \). The graph will have the general shape of a cubic curve (S - shaped with one inflection point) consistent with a positive leading - coefficient cubic. If the options include a cubic - shaped graph with this end - behavior and the appropriate curvature (from the expanded form's coefficients), that graph is the correct one. (Since the specific graph options are not fully detailed in the provided text, the above is the method to identify the correct graph.)