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Sum, Difference and Product Functions

Sum Functions

Given two functions f(x) and g(x), we can produce a new graph by adding them together: $y = f(x) + g(x)$.

  • This is sometimes written as: $y = (f + g)(x)$

To obtain the graph of the new function, we use a process called addition of ordinates.

  • This involves adding the y-values of the two original graphs for a sequence of x-values to gain a series of points that can then be joined with a smooth curve.

The sum function will only be defined for x values where both f(x) and g(x) are defined.

So the domain of the sum function will be:
… … $domain(f + g) = domain(f) \; \cap \; domain(g)$

Difference Functions

A difference function is the result of subtracting two functions: $y = f(x) - g(x)$

  • This is sometimes written as: $y = (f - g)(x)$

To obtain the graph, we adapt the addition of ordinates process by subtracting the y-values of the two original graphs for a sequence of x-values.

  • Or we could make the second graph negative and then add.

The difference function will only be defined where both f(x) and g(x) are defined.

So the domain of the difference function will be:
… … $domain(f - g) = domain(f) \; \cap \; domain(g)$

Product Functions

A product function is the result of multiplying two functions $y = f(x) \times g(x)$

  • This is sometimes written as: $y = (fg)(x)$

To obtain the graph, we adapt the addition of ordinates process by multiplying the y-values of the two original graphs for a sequence of x-values.

Note: Product functions can often be quite complicated so it is often best to use technology for these.

The product function will only be defined where both f(x) and g(x) are defined.

So the domain of the product function will be:
… … $domain(fg) = domain(f) \; \cap \; domain(g)$

Summary of Steps

For all three types of graph, the steps to follow are the same:

  1. Sketch the two original graphs in pencil or a light colour
  2. Draw in and label any asymptotes
  3. For a variety of x-values, produce the new y-coordinates by adding (or subtracting or multiplying) the original y-coordinates and mark each new point.
  4. Pay particular attention to:
    1. values close to any vertical asymptote(s)
    2. each end of the x-axis
    3. y-intercept
    4. x-intercepts of either original graph
    5. placement of x-intercepts and turning points
  5. Sketch the resultant graph by drawing a smooth curve through the points.
  6. Take care to draw the graph approaching the asymptotes (not touching or curling away)
  7. Label graphs, axes, asymptotes, intercepts, turning points,
  8. State the domain, range and the equations of asymptotes

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