When to use approxfun vs. approx - r

Question

The documentation for approxfun states that it is "often more useful than approx". I'm struggling to get my head around approxfun. When would approxfun be more useful than approx (and when would approx be more useful)?

Answer 1

approx returns the value of the approximated function at (either) specified points or at a given number of points. approxfun returns a function which can then be evaluated at some specific points. If you need the approximation at points that you know at the time of making the approximation, approx will do that for you. If you need a function (in the mathematical sense) which will return the value of the approximation for some argument given later, approxfun is what you need.
Here are some examples.
dat <- data.frame(x=1:10, y=(1:10)^2)
The output from approx and approxfun using this data
> approx(dat$x, dat$y)
$x
[1] 1.000000 1.183673 1.367347 1.551020 1.734694 1.918367 2.102041
[8] 2.285714 2.469388 2.653061 2.836735 3.020408 3.204082 3.387755
[15] 3.571429 3.755102 3.938776 4.122449 4.306122 4.489796 4.673469
[22] 4.857143 5.040816 5.224490 5.408163 5.591837 5.775510 5.959184
[29] 6.142857 6.326531 6.510204 6.693878 6.877551 7.061224 7.244898
[36] 7.428571 7.612245 7.795918 7.979592 8.163265 8.346939 8.530612
[43] 8.714286 8.897959 9.081633 9.265306 9.448980 9.632653 9.816327
[50] 10.000000
$y
[1] 1.000000 1.551020 2.102041 2.653061 3.204082 3.755102
[7] 4.510204 5.428571 6.346939 7.265306 8.183673 9.142857
[13] 10.428571 11.714286 13.000000 14.285714 15.571429 17.102041
[19] 18.755102 20.408163 22.061224 23.714286 25.448980 27.469388
[25] 29.489796 31.510204 33.530612 35.551020 37.857143 40.244898
[31] 42.632653 45.020408 47.408163 49.918367 52.673469 55.428571
[37] 58.183673 60.938776 63.693878 66.775510 69.897959 73.020408
[43] 76.142857 79.265306 82.551020 86.040816 89.530612 93.020408
[49] 96.510204 100.000000
> approxfun(dat$x, dat$y)
function (v)
.C(C_R_approxfun, as.double(x), as.double(y), as.integer(n),
xout = as.double(v), as.integer(length(v)), as.integer(method),
as.double(yleft), as.double(yright), as.double(f), NAOK = TRUE,
PACKAGE = "stats")$xout
<bytecode: 0x05244854>
<environment: 0x030632fc>
More examples of usage:
a <- approx(dat$x, dat$y)
af <- approxfun(dat$x, dat$y)
plot(dat)
points(a, pch=2)
plot(dat)
curve(af, add=TRUE)
or another example where a function is needed:
> uniroot(function(x) {af(x)-4}, interval=c(1,10))
$root
[1] 1.999994
$f.root
[1] -1.736297e-05
$iter
[1] 24
$estim.prec
[1] 6.103516e-05