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:
Given functions:
[tex]\begin{cases}f(x)=-4^x+5\\g(x)=x^3+x^2-4x+5 \end{cases}[/tex]
Function f(x)
This is an exponential function.
An exponential function includes a real number with an exponent containing a variable.
x-intercept (when y = 0):
[tex]\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}[/tex]
Therefore, the x-intercept of f(x) is (1.16, 0).
y-intercept (when x = 0):
[tex]\begin{aligned}f(0) & = -4^{0}+5\\& = 1+5\\& = 6\end{aligned}[/tex]
Therefore, the y-intercept of f(x) is (0, 6).
End behavior
[tex]\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty[/tex]
[tex]\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow 5[/tex]
Therefore, there is a horizontal asymptote at y = 5 which means the curve gets close to y = 5 but never touches it. Therefore:
- Domain: (-∞, ∞)
- Range: (-∞, 5)
Function g(x)
This is a polynomial function of degree 3 (since the greatest exponent of the function is 3).
A polynomial function is made up of variables, constants and exponents that are combined using mathematical operations.
x-intercept (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.
y-intercept (when x = 0):
[tex]\begin{aligned}g(0) & = (0)^3+(0)^2-4(0)+5\\& = 0+0+0+5\\& = 5 \end{aligned}[/tex]
Therefore, the y-intercept of g(x) is (0, 5).
End behavior
[tex]\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty[/tex]
[tex]\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow - \infty[/tex]
Therefore:
- Domain: (-∞, ∞)
- Range: (-∞, ∞)
Conclusion
Key features both functions have in common:
- One x-intercept (though not the same)
- One y-intercept (though not the same)
- Same unrestricted domain: (-∞, ∞)
