{"id":4174,"date":"2010-02-22T21:50:20","date_gmt":"2010-02-22T13:50:20","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=4174"},"modified":"2010-02-26T09:07:51","modified_gmt":"2010-02-26T01:07:51","slug":"functions","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/functions-4174","title":{"rendered":"Functions"},"content":{"rendered":"

A lot of people have difficulty with functions<\/strong> in math. I suspect it's because function notation is not very learner-friendly.<\/p>\n

Yousuf, one of my regular correspondents, got stuck on the following problem recently.<\/p>\n

What is the area of the rectangle ADEB shown in the diagram?<\/b><\/p>\n

The curve is the graph of y<\/em> = 1\/x<\/em>2<\/sup> (for positive x<\/em>), and r<\/em> is some arbitrary value of x<\/em>.<\/p>\n

\"rectangle\" <\/p>\n

We'll come back to this question a little later. I suspect his problem with this question was due to a rusty conceptual understanding of functions.<\/p>\n

Functions Overview<\/h2>\n

A function is simply an expression involving variable(s). <\/p>\n

We usually write a function of the variable x<\/em> using the notation: f<\/em>(x<\/em>). A function has at most 1 value for each value of x<\/em>.<\/p>\n

For example, if f<\/em>(x<\/em>) = 5x<\/em>2<\/sup> + 3, we can find the value of the function if we choose x <\/em>= 0 as follows.<\/p>\n

f<\/em>(0) = 5(0)2<\/sup> + 3 = 5 × 0 + 3 = 3 <\/p>\n

Now, this is a good example of the notation<\/strong> problem I was talking about at the beginning. We write "f<\/em>(0)" (f<\/em> bracket 0 bracket) to mean "evaluate the function expression by substuting 0 every time we see an x<\/em>" and we see this on the left hand side of this equation.<\/p>\n

But on the right hand side, I have written "5(0)2<\/sup>" (5 bracket 0 bracket squared) and this means "5 × 02<\/sup>". We need to be careful with this - writing 2 different concepts with what is essentially the same notation. <\/p>\n

It is a shame that function notation is so clumsy and causes problems for newbies. <\/p>\n

Let's look at some more examples for our function f<\/em>(x<\/em>) = 5x<\/em>2<\/sup> + 3. <\/p>\n

f<\/em>(2) = 5(2)2<\/sup> + 3 = 5 × 4 + 3 = 23.<\/p>\n

f<\/em>(10) = 5(10)2<\/sup> + 3 = 5 × 100 + 3 = 503.<\/p>\n

If we were to substitute many more values of x<\/em> and plot the dots on a graph, we would get the following:<\/p>\n

\"x^2 <\/p>\n

Note: <\/strong>On the vertical axis, I put f<\/em>(x<\/em>), but I could have also put "y<\/em>", since the convention in math is the vertical axis represents the function value. Often you'll see it written y = f<\/em>(x<\/em>).<\/p>\n

OK so far?<\/p>\n

Now, let's make things a bit more interesting. What is f<\/em>(a<\/em>)? We just substitute a<\/em> everywhere there is an x<\/em> in the original function, like we did before with the numbers:<\/p>\n

f<\/em>(a<\/em>) = 5(a<\/em>)2<\/sup> + 3 = 5a<\/em>2<\/sup> + 3<\/p>\n

Let's do another. In this next case, f<\/em>(a<\/em> + 4), we are just replacing each x<\/em> in the original function expression with a<\/em> + 4. <\/p>\n

f<\/em>(a<\/em> + 4) = 5(a<\/em> + 4)2<\/sup> + 3 = 5(a<\/em>2<\/sup> + 8a<\/em> + 16) + 3 = 5a<\/em>2<\/sup> + 40a<\/em> + 83<\/p>\n

Of course, we need to be careful to expand out the brackets properly!<\/p>\n

A Different Function<\/h2>\n

Let's change our function to \"1\/(x^2)\" . <\/p>\n

This is the curve we met in the question at the the beginning of this article. <\/p>\n

If x<\/em> = 1, <\/em>we replace every x<\/em> in our expression with 1 and we have:<\/p>\n

\"f(1)\"<\/p>\n

What f<\/em>(1) means on a graph is the distance from the x-<\/em>axis to the graph is 1 unit. The function value is the height<\/strong> of the graph for that x<\/em>-value.<\/p>\n

\"AB\"<\/p>\n

Now let's do f<\/em>(3a<\/em>).<\/p>\n

\"f(3a)\" <\/p>\n

The value <\/p>\n

\"1 <\/p>\n

represents the height of the graph when x<\/em> = 3a.<\/em> We need to be careful with the brackets. <\/p>\n

Back to Our Problem <\/h2>\n

Here's the graph again:<\/p>\n

\"rectangle\"<\/p>\n

So how do we find the area of the rectangle BADE? The width <\/strong>of the rectangle is quite straightforward, as the distance from r<\/em> to r<\/em> − 1 is just 1 unit. But we need to find the height AD. <\/p>\n

AD is just the function value f<\/em>(r<\/em>):<\/p>\n

\"f(r)\"<\/p>\n

So the area of the rectangle is just <\/p>\n

Area = width × height = \"1\/r^2\"<\/p>\n

What if we needed the height BC?<\/h2>\n

We would just find the function value as follows.<\/p>\n

BC = \"f(r<\/p>\n

Functions of 2 Variables<\/h2>\n

The functions above only have one variable and they describe a curve in 2-D space.<\/p>\n

To describe a 3-D surface, we need to use 2 variables. <\/p>\n

We write a function of 2 variables using this notation:<\/p>\n

z = f<\/em>(x,y<\/em>)<\/p>\n

The "z<\/em>" indicates the height of the surface for particular values of x<\/em> and y<\/em>.<\/p>\n

An example of a 3-dimensional surface is z<\/em> = x<\/em>2<\/sup> + 3 sin y.<\/em><\/p>\n

\"x^2<\/em><\/p>\n

More Information <\/h2>\n

See this chapter for a lot more examples of functions: Functions and Graphs<\/a>. (2 dimensional)<\/p>\n

This is an introduction to 3-dimensional Coordinate System.<\/a><\/p>\n

See also Towards more meaningful math notation<\/a> where I suggest an alternative to the current confusion. <\/p>\n

See the 14 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

\"functions\"<\/a>The concept of functions causes a lot of confusion. This article attempts to make things a bit clearer.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[134],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4174"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=4174"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4174\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=4174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=4174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=4174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}