How to find graph of the sum of two functions

Question

Suppose I know the graphs of two functions $f(x)$ and $g(x)$. How can I find the graph of $h(x)=f(x)+g(x)$? What are the rules to be followed ?

P.S. In case my question seems silly,at least provide me with a link or something so that I can learn!

Answer 1

Since $h(x)=(f+g)(x):=f(x)+g(x)$ for every $x$ in the domain, the graph is the one that you obtain summing the two functions pointwise.

That is, at $x=x_0$ will correspond the point $h(x_0)=f(x_0)+g(x_0)$.

Edited after seeing the comment about discontinuities: if one of the functions $f$ and $g$ has a discontinuity, remember that the domain of $f+g$ is $\mathcal {D}_{f+g}=\mathcal{D}_f \cap \mathcal{D}_g$. You can only sum the two functions where they both exists and in these points the same logic applies.

Answer 2

You can think about the graph of $h(x)$ pointwise, adding the heights of the two graphs $f(x)$ and $g(x)$ at each point $x$. For example, if $f(1) = 2$ and $g(1) = 3$. $$h(1) = f(1) + g(1) = 2 + 3 = 5$$

Everything is all good now.

Answer 3

You need to do some analysis.I recommend take the following points.

From basic function f and g:

1. See when are f and g zero

2. Find the max and min value of the f and g (example : for sin(x) +1 and -1)

3. Plot the envelopes of the shape in the enlarged size to get an idea of the graph.

   OR

4. Calculate the roots of function(sum) if possible.

5. Analyse the value at the roots.

6. Find the differential and analyze the differentiability

7. Find local maxima and minimas and on the basis of differentiability plot the curve.

8. You may want to analyze the concavity and convexity.For that,find the double differential.

You may like to watch this video. Curve Sketching