Vector Space of Continuous Real Valued Functions

What type of function has a range that does not consist of numbers?

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

A function whose domain is a set of real numbers and whose range is a set of vectors is considered a vector function, sometimes called a vector valued function.

Terminology

But before we can talk about vector functions and space curves, we must take a step back and review some terminology.

A relation is a pairing of quantities, like the number of gallons of gas, the amount of money needed to purchase it, or the pairing of \(x\) and \(y\) coordinates to form an ordered pair \((x,y)\).

And a function is special type of relation in which for every input value there is exactly one output value.

The set of inputs is called the domain and the set of outputs is called the codomain.

It is important to note that we refer to the codomain as the range when the possible and actual outcomes are the same.

For our purposes, we will simply say the domain are the inputs and the range are the outputs to keep things straightforward.

Quick Example

Let's ensure we know our definitions from above by quickly walking through an example.

For instance, the relation \(\left\{ {\left( {2, – 4} \right),\left( { – 3,7} \right),\left( {5,1} \right),\left( {0,13} \right)} \right\}\) is a function because each input value is unique. And we can say that the domain is the set of first elements\(\left\{ {2, – 3,5,0} \right\}\) and the range, or codomain, is the set of second elements \(\left\{ { – 4,7,1,13} \right\}\)

Easy enough to understand, right?

Introduction To Vector Functions

So, now let's get back to vector functions and space curves.

Definition

A vector function is a function where each real number in the domain is mapped to either a two- or three-dimensional vector, which allows us to locate where in space a particular object is at any time.

If \(\color{#BF401D} f\left( t \right)\), \(\color{#1D7329}g\left( t \right)\), and \(\color{#1394BF}h\left( t \right)\) are the components of the vector \(\overrightarrow {r\left( t \right)} \), then \(f\), \(g\), and \(h\) are real-valued functions called the component functions of vector \(\overrightarrow {r\left( t \right)} \) and \(t\) is the independent variable or time, and is denoted

\begin{equation}
\overrightarrow{r(t)}=\langle \color{#BF401D}f(t), \color{#1D7329}g(t), \color{#1394BF}h(t)\rangle
\end{equation}

Example

For instance, if \(\overrightarrow {r\left( t \right)} = \left\langle {\sqrt t ,{t^2},\ln \left( {t – 1} \right)} \right\rangle \), then the component functions are \(\color{#BF401D} f\left( t \right) = \sqrt t \), \(\color{#1D7329}g\left( t \right) = {t^2}\) and \(\color{#1394BF}h\left( t \right) = \ln \left( {t – 1} \right)\), and domain of \(\overrightarrow {r\left( t \right)} \) consists of all values of \(t\) for which \(\overrightarrow {r\left( t \right)} \) is defined.

Therefore, to determine the appropriate domain for this vector-valued function we must investigate each component separately.

• \(\sqrt{t}\) is valid only when \(t \ge 0\)
• \({t^2}\) is valid for all real numbers
• \(\ln \left( {t – 1} \right)\) is valid only when \(t>1\)

Consequently, the domain of \(\overrightarrow {r\left( t \right)} \) is the interval \(\left( {1,\infty } \right)\) where all three component functions are valid simultaneously.

Don't worry.

We will walk through numerous examples of how to represent a vector-valued function and determine its domain in the video lesson below.

Limit Of A Vector Function

But there's more to vector-valued functions than just finding domain and range.

Did you know that to find the limit of a vector function, all we have to do is take the limit of its component functions, provided the limits of the component function exist?

If \(\overrightarrow {r\left( t \right)} = \left\langle {\color{#BF401D} f\left( t \right),\color{#1D7329}g\left( t \right),\color{#1394BF}h\left( t \right)} \right\rangle \), then \(\mathop {\lim }\limits_{t \to a} \overrightarrow {r\left( t \right)} = \left\langle {\mathop {\lim }\limits_{t \to a} {\color{#BF401D} f\left( t \right)},\mathop {\lim }\limits_{t \to a} {\color{#1D7329}g\left( t \right)},\mathop {\lim }\limits_{t \to a} \color{#1394BF}h\left( t \right)} \right\rangle \)

For example, if \(\overrightarrow {r\left( t \right)} = \left\langle {\sqrt t ,{t^2},\ln \left( {t – 1} \right)} \right\rangle \), then let's find \(\mathop {\lim }\limits_{t \to 4} \overrightarrow {r\left( t \right)} \)

\begin{equation}
\begin{gathered}
\lim _{t \rightarrow 4} \overrightarrow{r(t)}=\left\langle\lim _{t \rightarrow 4} \sqrt{t}, \lim _{t \rightarrow 4} t^{2}, \lim _{t \rightarrow 4} \ln (t-1)\right\rangle \\
=\left\langle\sqrt{4}, 4^{2}, \ln (4-1)\right\rangle \\
=\langle 2,16, \ln 3\rangle
\end{gathered}
\end{equation}

And what's cool is that the limit of vector-valued function obeys the same rules as limits of real-valued functions, so we can use all of our old techniques (even L'Hospital's rule) to help us evaluate!

Space Curve

Now, a vector function is continuous if and only if its component functions \(f\), \(g\), and \(h\) are also continuous.

And supposing \(f\), \(g\), and \(h\) are continuous real-valued functions, then the set \(C\) of all points \((x,y,z)\) in space, where \(x = \color{#BF401D} f\left( t \right)\), \(y = \color{#1D7329}g\left( t \right)\), and \(z = \color{#1394BF}h\left( t \right)\) are parametric equations of \(C\) and \(t\) is the parameter, is called a space curve.

In other words, a space curve is a curve in space and is defined by a vector function.

Graphing Vector Valued Functions

Now we know that a vector consists of both direction and magnitude, and it is considered to be in standard position if the initial point is located at the origin.

Thus, when graphing a vector-valued function we graph the vectors in standard position such that the result is a:

• Plane curve when \(\overrightarrow {r\left( t \right)} = \left\langle {\color{#BF401D} f\left( t \right),\color{#1D7329}g\left( t \right)} \right\rangle \) is a two-dimensional vector and consists of the set of all points \(\left( {t,\overrightarrow {r\left( t \right)} } \right)\).
• Space curve when is a three-dimensional vector consisting of the set of all points \(\left( {t,\overrightarrow {r\left( t \right)} } \right)\).

Note that graphing a plane curve can be readily done by hand, but we tend to utilize a computer to assist us in graphing space curves because a plane curve lies in a single plane, whereas a space curve may pass through any region of a 3-space.

Plane Curve Vs. Space Curve

Furthermore, it should be noted that any representation of a plane curve or space curve is called a vector parameterization of the curve.

Together we will learn how to represent vector functions and determine their domain. Additionally, we will learn how to evaluate the limit of a vector function, determine continuity, and even sketch the graphs of both plane curves and space curves.

Let's jump right in.

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

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Source: https://calcworkshop.com/vector-functions/vector-valued-function/