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Combine function rules and ordinates

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Combining functions

A combined function uses the component values at the same input and then applies the stated operation. Its domain is restricted to inputs where the required component values exist

\[h(x)=f(x)-g(x),\quad D_h=D_f\cap D_g\]
  1. 1Use the common domain A sum, difference or product is defined only where both component functions are defined. A square of one function keeps that function's domain \(\displaystyle D_{f+g}=D_{f-g}=D_{fg}=D_f\cap D_g,\quad D_{f^2}=D_f\)
  2. 2Combine matching ordinates At the same input, apply the stated operation to the two component values \(\displaystyle f(3)=7,\quad g(3)=2\quad\Rightarrow\quad (f-g)(3)=7-2=5\)
  3. 3Expand each polynomial term For polynomial rules, distribute the operation across like powers and then collect like terms \(\displaystyle (2x+1)+(x-3)=3x-2\)
  4. 4Check zeros and signs A product is zero when at least one factor is zero, and a squared ordinate is never negative \(\displaystyle f(x)g(x)=0\text{ if }f(x)=0\text{ or }g(x)=0,\quad f(x)^2\ge0\)
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Combining functions

A combined function uses the component values at the same input and then applies the stated operation. Its domain is restricted to inputs where the required component values exist

\[h(x)=f(x)-g(x),\quad D_h=D_f\cap D_g\]
  1. Use the common domain A sum, difference or product is defined only where both component functions are defined. A square of one function keeps that function's domain \(\displaystyle D_{f+g}=D_{f-g}=D_{fg}=D_f\cap D_g,\quad D_{f^2}=D_f\)
  2. Combine matching ordinates At the same input, apply the stated operation to the two component values \(\displaystyle f(3)=7,\quad g(3)=2\quad\Rightarrow\quad (f-g)(3)=7-2=5\)
  3. Expand each polynomial term For polynomial rules, distribute the operation across like powers and then collect like terms \(\displaystyle (2x+1)+(x-3)=3x-2\)
  4. Check zeros and signs A product is zero when at least one factor is zero, and a squared ordinate is never negative \(\displaystyle f(x)g(x)=0\text{ if }f(x)=0\text{ or }g(x)=0,\quad f(x)^2\ge0\)

Worked example

Let the two functions be defined on the intervals shown. Form their difference and find its value at the given input

\[f(x)=3x-2,\quad x\in\left[-1,6\right],\qquad g(x)=x+4,\quad x\in\left[2,8\right],\qquad h(x)=f(x)-g(x)\]
  1. \(\displaystyle D_h=D_f\cap D_g=\left[2,6\right]\)Both component rules must be defined at an input used by h.
  2. \(\displaystyle h(x)=(3x-2)-(x+4)\)Substitute the two component rules into the difference
  3. \(\displaystyle h(x)=3x-2-x-4=2x-6\)Distribute the subtraction and collect like terms
  4. \(\displaystyle h(5)=2(5)-6=4\)The input 5 is in the common domain, so substitute it into h.

Common mistakes

Using the union for the domain
✕\(\displaystyle D_f=\left[-2,6\right],\quad D_g=\left[1,8\right]\quad\Rightarrow\quad D_h=\left[-2,8\right]\)✓\(\displaystyle D_f=\left[-2,6\right],\quad D_g=\left[1,8\right]\quad\Rightarrow\quad D_h=\left[1,6\right]\)
A combined value requires both component values, so the domains must be intersected
Using the wrong operation
✕\(\displaystyle f(5)=11,\quad g(5)=3\quad\Rightarrow\quad h(5)=11+3=14\)✓\(\displaystyle f(5)=11,\quad g(5)=3\quad\Rightarrow\quad h(5)=11-3=8\)
For a difference, subtract the second ordinate from the first rather than adding the two ordinates
Keeping a negative square
✕\(\displaystyle f(-5)=-4\quad\Rightarrow\quad f(-5)^2=-16\)✓\(\displaystyle f(-5)=-4\quad\Rightarrow\quad f(-5)^2=16\)
Squaring a negative ordinate gives a positive result

Difficulty levels

introCombine a short table of exact ordinates using a stated operation
standardForm a polynomial sum, difference or product rule and state its intersection domain
challengeUse exact ordinate data to identify zeros, signs, shared points or features of a square
examSelect a combined graph description and verify one exact ordinate

Questions about this topic

Why is a sum's domain an intersection?+

Both functions must have values at the same input, so only inputs in both component domains can be used

How do I find the difference of two functions at an input?+

Evaluate both functions at that input, then subtract the second value from the first. The input must belong to both component domains

Why is a product zero when one factor is zero?+

Multiplication by zero gives zero, so the product is zero whenever either function has value zero at that input

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs

In this chapter

Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model