Combining Functions

There are many different ways to combine functions. Let us use P(x) and Q(x) as examples.

P(x)=x+3 Q(x)= 2x-5

Types of Combinations

  1. (P+Q)(x) or P(x)+Q(x)

  2. (P-Q)(x) or P(x)-Q(x)

  3. (PQ)(x) or P(x) times Q(x)

  4. (P/Q)(x) or P(x) divided by Q(x)

  5. (P 0 Q)(x) or P(Q(x)) or P of Q of x

  6. (Q 0 P)(x) or Q(P(x)) or Q of P of x

How to Use

First, recall how normal functions work. f(x) is like saying Y, because it is saying when x performs this function, it is equal to Y. The same applies here, P and Q, are like their own y values. When x performs this function, it is equal to P/Q etc. To combine functions, Set the functions (P+Q)(x) etc. equal to their operations. Remember P(x)=x+3 and Q(x)=2x-5.

  1. First set up (P+Q)(x)=(x+3)+(2x-5). Now we have an equation that can be simplified. Answer is (P+Q)(x)=-2+3x

  2. The same idea follows for the rest. (P-Q)(x)=(x+3)-(2x-5) → (P-Q)(x)=8-x. Note that the negative sign is applied to the entire second function.

  3. (PQ)(x)=(x+3)(2x-5) → (PQ)(x)=2x²+x-15. Note that we now have a solvable quadratic.

  4. (P/Q)(x)=(x+3)/(2x-5) → (P/Q)(x)=(x+3)/(2x-5). This equation is already fully simplified.

  5. Now we see something different. The 0 in this equation represents that we are using one function to plug in for x of another function. So (P 0 Q)(x) or P(Q(x))=(2x-5)+3 → P(Q(x))=2x-2. See how we plugged in the entire Q(x) equation for x.

  6. (Q 0 P)(x)=2(x+3)-5 → 2x+1. We plugged in the entire P(x) equation for Q.