Return to Revisions

To compute the intersection(s) of the graphs of the exponential function 'y=e^x' and the line 'y=mx+b' (as already noted in the first answer to this question), it is convenient to consider the function 'f(x)=e^x-mx-b'. Since 'f' is strictly convex, it has at most two roots. Next, depending on the sign of 'm', we determine how many roots are possible and, if there are two, we bracket them. Then the roots are computed using the bisection method. This is exactly the approach implemented in the code below. I added two more examples to illustrate an important point: the roots exist, but they lie outside our interval.

\documentclass[tikz,border=1cm]{standalone}
\usepackage{luacode}

\begin{luacode*}
-- Compute intersections of y=exp(x) with y=m*x+b on [xmin,xmax]
-- Export x-roots into TeX macro \roots (empty if none).
function lineExpIntersections(m,b,xmin,xmax)
  local function f(x) return math.exp(x) - (m*x + b) end

  local function bisect(L,R)
    local fL, fR = f(L), f(R)
    if fL == 0 then return L end
    if fR == 0 then return R end
    if fL*fR > 0 then return nil end
    for _=1,80 do
      local M  = 0.5*(L+R)
      local fM = f(M)
      if fM == 0 then return M end
      if fL*fM < 0 then
        R, fR = M, fM
      else
        L, fL = M, fM
      end
    end
    return 0.5*(L+R)
  end

  local roots = {}

  if m < 0 then
    -- strictly increasing => at most one root; find it if bracketed
    local r = bisect(xmin, xmax)
    if r then roots[#roots+1] = r end

  elseif m == 0 then
    -- exp(x)=b
    if b > 0 then
      local r = math.log(b)
      if r >= xmin and r <= xmax then roots[#roots+1] = r end
    end

  else
    -- m>0: 0/1/2 roots; minimum at x0=log(m)
    local x0   = math.log(m)
    local fmin = m - (m*math.log(m) + b) -- f(x0)

    if fmin == 0 then
      if x0 >= xmin and x0 <= xmax then roots[#roots+1] = x0 end
    elseif fmin < 0 then
      -- two roots globally; try to bracket each inside [xmin,xmax]
      local L1, R1 = xmin, math.min(x0, xmax)
      local L2, R2 = math.max(x0, xmin), xmax
      local r1 = bisect(L1, R1)
      local r2 = bisect(L2, R2)
      if r1 then roots[#roots+1] = r1 end
      if r2 then roots[#roots+1] = r2 end
    end
  end

  local out = {}
  for i=1,#roots do out[i] = string.format("%.12f", roots[i]) end
  tex.sprint("\\def\\roots{" .. table.concat(out, ",") .. "}")
end
\end{luacode*}

% TeX wrapper: defines \roots
\newcommand{\LineExpIntersections}[4]{%
  \directlua{lineExpIntersections(#1,#2,#3,#4)}%
}

\begin{document}
\begin{tikzpicture}
  \draw[->] (-2,0) -- (2,0);
  \draw[->] (0,-2) -- (0,{exp(2)});
  \draw[domain=-2:2,samples=200] plot (\x,{exp(\x)});

  \def\xmin{-2}
  \def\xmax{2}

  % 1) y = x + 1.5 (two intersections)
  \draw[domain=\xmin:\xmax,samples=2] plot (\x,{1*\x+1.5});
  \LineExpIntersections{1}{1.5}{\xmin}{\xmax}
  \foreach \r in \roots { \fill[red] (\r,{exp(\r)}) circle (1.2pt); }

  % 2) y = x + 1 (one intersection)
  \draw[domain=\xmin:\xmax,samples=2] plot (\x,{1*\x+1});
  \LineExpIntersections{1}{1}{\xmin}{\xmax}
  \foreach \r in \roots { \fill[green] (\r,{exp(\r)}) circle (1.2pt); }

  % 3) y = -x + 4.5 (one intersection)
  \draw[domain=\xmin:\xmax,samples=2] plot (\x,{-1*\x+4.5});
  \LineExpIntersections{-1}{4.5}{\xmin}{\xmax}
  \foreach \r in \roots { \fill[blue] (\r,{exp(\r)}) circle (1.2pt); }

  % 4) y = x (zero intersections)
  \draw[domain=\xmin:\xmax,samples=2] plot (\x,{1*\x});
  \LineExpIntersections{1}{0}{\xmin}{\xmax}
  \foreach \r in \roots { \fill (\r,{exp(\r)}) circle (1.2pt); }
  
  % 5) y = x + 4 (zero intersections)
  \draw[domain=\xmin:\xmax,samples=2] plot (\x,{1*\x+4});
  \LineExpIntersections{1}{4}{\xmin}{\xmax}
  \foreach \r in \roots { \fill[orange] (\r,{exp(\r)}) circle (1.2pt); }
  
  % 6) y = x + 5 (no intersections on [xmin,xmax]” или “roots outside the interval)  
  \def\txmax{1.5}
  \draw[domain=\xmin:\txmax,samples=2] plot (\x,{1*\x+5});
  \LineExpIntersections{1}{5}{\xmin}{\txmax}
  \foreach \r in \roots { \fill[orange] (\r,{exp(\r)}) circle (1.2pt); }

\end{tikzpicture}
\end{document}

P.S. If 'm' or 'fmin' is obtained as the result of computations, an “exactly zero” comparison may fail. In our examples everything is fine, but in the general case one should use something like 'if math.abs(m) < eps then ...', and so on.