Question 8 (Essay Worth 10 points)
(07.04 MC)
Given the functions $f(x)=x^{3}+x^{2}-3 x+4$ 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
\(\boxed{\text{Final Answer: The function } f(x) \text{ is a cubic function and the function } g(x) \text{ is an exponential function. The domain of both functions is all real numbers. The range of } f(x) \text{ is all real numbers and the range of } g(x) \text{ is } (-4, \infty). \text{ The x-intercepts of } f(x) \text{ are approximately } [-1.65063, 0.325316 - 1.31443i, 0.325316 + 1.31443i] \text{ and the x-intercept of } g(x) \text{ is } 2. \text{ The y-intercepts of } f(x) \text{ and } g(x) \text{ are } 4 \text{ and } -4 \text{ respectively. The key feature that } f(x) \text{ and } g(x) \text{ have in common is that they both have a y-intercept at } y=4 \text{ and } y=-4 \text{ respectively.}}
Step 1 ：The function \(f(x)\) is a polynomial function of degree 3, also known as a cubic function. This is because it is composed of terms with non-negative integer exponents and the highest exponent is 3.
Step 2 ：The function \(g(x)\) is an exponential function. This is because it involves an exponent of x.
Step 3 ：The domain of both functions is all real numbers. This is because there are no restrictions on the values that x can take in either function.
Step 4 ：The range of \(f(x)\) is all real numbers. This is because a cubic function can take any real value depending on the value of x.
Step 5 ：The range of \(g(x)\) is \((-4,∞)\). This is because the base of the exponential function is positive (2), so the function is always positive, but we subtract 4, so the minimum value is -4.
Step 6 ：The x-intercepts of a function are the values of x for which the function equals zero. For \(f(x)\), we would need to solve the equation \(x^{3}+x^{2}-3x+4=0\). For \(g(x)\), we would need to solve the equation \(2^{x}-4=0\).
Step 7 ：The y-intercepts of a function are the values of the function when x equals zero. For \(f(x)\), this would be \(f(0)=4\). For \(g(x)\), this would be \(g(0)=-4\).
Step 8 ：The key feature that \(f(x)\) and \(g(x)\) have in common is that they both have a y-intercept at y=4 and y=-4 respectively.
Step 9 ：The x-intercepts of \(f(x)\) are approximately \([-1.65063, 0.325316 - 1.31443i, 0.325316 + 1.31443i]\) and the x-intercept of \(g(x)\) is \(2\).
Step 10 ：\(\boxed{\text{Final Answer: The function } f(x) \text{ is a cubic function and the function } g(x) \text{ is an exponential function. The domain of both functions is all real numbers. The range of } f(x) \text{ is all real numbers and the range of } g(x) \text{ is } (-4, \infty). \text{ The x-intercepts of } f(x) \text{ are approximately } [-1.65063, 0.325316 - 1.31443i, 0.325316 + 1.31443i] \text{ and the x-intercept of } g(x) \text{ is } 2. \text{ The y-intercepts of } f(x) \text{ and } g(x) \text{ are } 4 \text{ and } -4 \text{ respectively. The key feature that } f(x) \text{ and } g(x) \text{ have in common is that they both have a y-intercept at } y=4 \text{ and } y=-4 \text{ respectively.}}
