Question

Given the functions f(x) = –4ˣ + 5 and g(x) = x³ + x² – 4x + 5, 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 1

• 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.

Explanation:

Given functions:

`\begin{cases}f(x)=-4^x+5\\g(x)=x^3+x^2-4x+5 \end{cases}`

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):

`\begin{aligned}f(x) & = 0\\\implies -4^x+5 & =0\\ 4^x &=5\\\ln 4^x &= \ln 5\\x \ln 4 &= \ln 5\\x&=( \ln 5)/(\ln 4)\\x&=1.16\:\: \sf(2\:d.p.)\end{aligned}`

Therefore, the x-intercept of f(x) is (1.16, 0).

y-intercept (when x = 0):

`\begin{aligned}f(0) & = -4^(0)+5\\& = 1+5\\& = 6\end{aligned}`

Therefore, the y-intercept of f(x) is (0, 6).

End behavior

`\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty`

`\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow 5`

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):

`\begin{aligned}g(0) & = (0)^3+(0)^2-4(0)+5\\& = 0+0+0+5\\& = 5 \end{aligned}`

Therefore, the y-intercept of g(x) is (0, 5).

End behavior

`\textsf{As }x \rightarrow \infty, \: f(x) \rightarrow \infty`

`\textsf{As }x \rightarrow -\infty, \: f(x) \rightarrow - \infty`

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: (-∞, ∞)