"How can I build up anonymous functions from other anonymous functions"
Just call the functions, exactly like you would any other type of function:
f_1 = @(x,y) erf(x+y)-erf(x-y);
f_2 = @(x,y) exp(x*y+2*x);
f = @(x,y) f_1(x,y)*f_2(x,y); % you need to call them!
"How can I get a matrix from inputting two vectors?"
That depends on which version of MATLAB you are using and what operations your code contains.
Method one: vectorize: If you are using a version >=R2016b then make sure that you vectorize the operations, e.g. using element-wise array operations instead of matrix operations:
f_2 = @(x,y) exp(x.*y+2*x);
% ^^ array operation!
and then call the function with one row vector and one column vector. For earlier versions you will instead need to use bsxfun for any operation that involves different sized arrays.
Method two: arrayfun: Otherwise you can generate the required matrices using ndgrid and use a loop or arrayfun:
>> f_1=@(x,y) (erf(x+y)-erf(x-y));
>> f_2=@(x,y) (exp(x*y+2*x));
>> f =@(x,y) (f_1(x,y)*f_2(x,y));
>> x = linspace(-2,2,10);
>> y = linspace(-2,2,10);
>> [xM,yM] = ndgrid(x,y);
>> zM = arrayfun(f,xM,yM)
zM =
-1 -0.21775 -0.035275 -0.0041123 -0.00029302 0.00012046 0.00028574 0.00041426 0.00043219 0.00033546
-1.4703 -0.50089 -0.13285 -0.026021 -0.0029847 0.001495 0.0032701 0.0041888 0.0039625 0.0029184
-1.7913 -0.89723 -0.37183 -0.11768 -0.020722 0.012646 0.026748 0.031478 0.028291 0.021036
-1.9405 -1.3307 -0.80635 -0.38671 -0.098104 0.072947 0.15898 0.18328 0.16723 0.13483
-1.9864 -1.7473 -1.4215 -0.9381 -0.31685 0.28705 0.69754 0.86751 0.87523 0.81663
-1.9864 -2.1289 -2.1102 -1.6967 -0.69822 0.7707 2.2819 3.4577 4.2503 4.8318
-1.9405 -2.4068 -2.6378 -2.2881 -1.0498 1.4119 5.5656 11.605 19.151 27.927
-1.7913 -2.409 -2.6804 -2.2777 -1.0768 1.7645 10.021 31.662 76.402 152.53
-1.4703 -1.9964 -2.1104 -1.6475 -0.75323 1.5038 13.11 66.931 252.36 740.79
-1 -1.2883 -1.2349 -0.85177 -0.35909 0.87347 12.259 105.15 649.09 2981
