Plot[Log[1 + Exp[x]], {x, -2, 2}]

a = 0.4;

Plot[{a Log[x], (1 - a) Log[1 - x], 
        a Log[x] + (1 - a) Log[1 - x]}, {x, 0.1,  .9}]

Fun

Plot[{a Log[x], (1 - a) Log[1 - x], 
        a Log[x] + (1 - a) Log[1 - x]}, {x, 0.1,  .9}]

Plot[
      Piecewise[{{1 - x, x < 1}, {0, x ≥ 1}}], {x, -2, 
        2}]

hinge1[x_] := 
      Piecewise[{{1 - x, x < 1}, {0, x ≥ 1}}]

hinge2[x_] := Piecewise[{{x, x > 0}, {0, x ≤ 0}}]

hinge[x_] := 
      Piecewise[{{1 - x, x < 1}, {0, x ≥ 1}}]

Plot[0.4  hinge1[x] + 0.3 hinge2[x], {x, -3, 3}]

Huber

Plot[{hinge1[x], hinge2[x]}, {x, -1, 1}]

g[x_] = Sin[x]

Plot[g[x], {x, 0, 1}]

f[x, t] = t

m = NIntegrate[f[x, #] + g[x] - f[x, #] g[x], {x, 0, 1}] &

m[1]