Answer:
- f(x) is an exponential function
- g(x) is a polynomial function of degree 3
- Key common features: same domain, both have one x-intercept and one y-intercept.
Step-by-step explanation:
<u>Given functions</u>:
![\begin{cases}f(x)=-4^x+5\\g(x)=x^3+x^2-4x+5 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Df%28x%29%3D-4%5Ex%2B5%5C%5Cg%28x%29%3Dx%5E3%2Bx%5E2-4x%2B5%20%5Cend%7Bcases%7D)
<h3><u>Function f(x)</u></h3>
This is an exponential function.
An exponential function includes a real number with an exponent containing a variable.
<u>x-intercept</u> (when y = 0):
![\begin{aligned}f(x) & = 0\\\implies -4^x+5 & =0\\ 4^x &=5\\\ln 4^x &= \ln 5\\x \ln 4 &= \ln 5\\x&=\dfrac{ \ln 5}{\ln 4}\\x&=1.16\:\: \sf(2\:d.p.)\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Df%28x%29%20%26%20%3D%200%5C%5C%5Cimplies%20-4%5Ex%2B5%20%26%20%3D0%5C%5C%204%5Ex%20%26%3D5%5C%5C%5Cln%204%5Ex%20%26%3D%20%5Cln%205%5C%5Cx%20%5Cln%204%20%26%3D%20%5Cln%205%5C%5Cx%26%3D%5Cdfrac%7B%20%5Cln%205%7D%7B%5Cln%204%7D%5C%5Cx%26%3D1.16%5C%3A%5C%3A%20%5Csf%282%5C%3Ad.p.%29%5Cend%7Baligned%7D)
Therefore, the x-intercept of f(x) is (1.16, 0).
<u>y-intercept</u> (when x = 0):
![\begin{aligned}f(0) & = -4^{0}+5\\& = 1+5\\& = 6\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Df%280%29%20%26%20%3D%20-4%5E%7B0%7D%2B5%5C%5C%26%20%3D%201%2B5%5C%5C%26%20%3D%206%5Cend%7Baligned%7D)
Therefore, the y-intercept of f(x) is (0, 6).
<u>End behavior</u>
![\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty](https://tex.z-dn.net/?f=%5Ctextsf%7BAs%20%7Dx%20%5Crightarrow%20%5Cinfty%2C%20%5C%3A%20f%28x%29%20%5Crightarrow%20%5Cinfty)
![\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow 5](https://tex.z-dn.net/?f=%5Ctextsf%7BAs%20%7Dx%20%5Crightarrow%20-%5Cinfty%2C%20%5C%3A%20f%28x%29%20%5Crightarrow%205)
Therefore, there is a <u>horizontal asymptote</u> at y = 5 which means the curve gets close to y = 5 but never touches it. Therefore:
- Domain: (-∞, ∞)
- Range: (-∞, 5)
<h3><u>Function g(x)</u></h3>
This is a polynomial function of degree 3 (since the greatest exponent of the function is 3).
A polynomial function is made up of <u>variables</u>, <u>constants</u> and <u>exponents</u> that are combined using mathematical operations.
<u>x-intercept</u> (when y = 0):
There is only one x-intercept of function g(x). It can be found algebraically using the Newton Raphson numerical method, or by using a calculator.
From a calculator, the x-intercept of g(x) is (-2.94, 0) to 2 decimal places.
<u>y-intercept</u> (when x = 0):
![\begin{aligned}g(0) & = (0)^3+(0)^2-4(0)+5\\& = 0+0+0+5\\& = 5 \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dg%280%29%20%26%20%3D%20%280%29%5E3%2B%280%29%5E2-4%280%29%2B5%5C%5C%26%20%3D%200%2B0%2B0%2B5%5C%5C%26%20%3D%205%20%5Cend%7Baligned%7D)
Therefore, the y-intercept of g(x) is (0, 5).
<u>End behavior</u>
![\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty](https://tex.z-dn.net/?f=%5Ctextsf%7BAs%20%7Dx%20%5Crightarrow%20%5Cinfty%2C%20%5C%3A%20f%28x%29%20%5Crightarrow%20%5Cinfty)
![\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow - \infty](https://tex.z-dn.net/?f=%5Ctextsf%7BAs%20%7Dx%20%5Crightarrow%20-%5Cinfty%2C%20%5C%3A%20f%28x%29%20%5Crightarrow%20-%20%5Cinfty)
Therefore:
- Domain: (-∞, ∞)
- Range: (-∞, ∞)
<u>Conclusion</u>
Key features both functions have in common:
- One x-intercept (though not the same)
- One y-intercept (though not the same)
- Same unrestricted domain: (-∞, ∞)