Question

Given the functions f(x) = x^3 + x^2 – 3x + 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 1

Hello there. To solve this question, we'll have to remember some properties about functions.

Given the functions:

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

We have to determine what type of functions are f(x) and g(x), as well as their key features and see which of them are common to both f and g.

First, f(x) is a polynomial function, as you can see you have a linear combination of x powers. In fact, it is a cubic function.

g, on the other hand, is an exponential function, as you can see by the term 2^x.

The key features of f, that is, domain, range, x and y-intercepts are:

Domain -

`(-\infty,\infty)`

Since it is continuous over all the real line.

Range -

`(-\infty,\infty)`

Because in fact f, as a polynomial function, is defined as:

`f:\mathbb{R}\rightarrow\mathbb{R}`

Of course, one can say that there are some functions that the range is not all the real numbers. A simple counterexample is in fact a quadratic function with vertex at (xv, yv). The range will be either defined as all the values above or below yv.

But when we're talking about cubic functions, we have in fact the range as the entire real line.

The x-intercept is given by plugging y = 0, that is, solving for the roots of f

`\begin{gathered} x^3+x^2-3x+4=0 \\ \end{gathered}`

In fact, we'll have a real solution (the only that interests us) and two conjugate complex solutions, approximately:

`x\approx-2.6780`

The y-intercept is given by plugging x = 0:

`f(0)=0^3+0^2-3\cdot0+4=4`

So y = 4 is the y-intercept of this cubic equation.

Its graph may looks like this:

Now, moving for g:

As said before, g is an exponential function.

So it is defined as:

`g:\mathbb{R}\rightarrow\mathbb{R}_(>-4)`

Of course, this definition for g is unique, since not all functions are defined like this for being exponential.

The domain, as you can see, is all the real numbers, since you can plug anything into 2^x - 4 and evaluate it to some number.

The range, as suggested by the definition given for g, is all the real numbers greater than -4, since 2^x is a power of 2, a positive number, it cannot be zero or negative no matter the choice of x made.

All exponential functions of this form have an asymptote at y = 0 (given the property of positive powers). But in this case g is translated 4 units down, so we say it has an asymptote at y = -4.

Therefore its domain is in fact the entire real line:

`(-\infty,\infty)`

And the range is

`(-4,\infty)`

The x-intercept is the roots of the function, that is

`2^x-4=0`

Add 4 on both sides of the equation

`2^x=4`

Take the base 2 logarithm on both sides of the equation

`\begin{gathered} \log_2(2^x)=\log_2(4) \\ x=2 \end{gathered}`

So the only real solution to this function is x = 2.

The y-intercept is found by plugging x = 0:

`g(0)=2^0-4=1-4=-3`

So we can finally say that the only key feature that f and g have in common is the domain.