# Function Arithmetic

When performing arithmetic on functions, it is necessary to understand the rules that relate functions to each other. For the following examples, we'll use these functions:

- \( a(x) = x^2 \)
- \( b(x) = x+3 \)
- \( c(x) = 4x \).

**Addition**: \( (f+g)(x) = f(x) + g(x) \), so for example,

\[ (b+c)(x) = b(x) + c(x) = (x+3) + (4x) = 5x + 3 .\]

**Subtraction**: \( (f-g)(x) = f(x) - g(x) \), so for example,

\[ (b-c)(x) = b(x) - c(x) = (x+3) - (4x) = -3x + 3. \]

**Multiplication**: \( ( f\cdot g)(x) = f(x) \cdot g(x) \), so for example,

\[ (a\cdot b)(x) = (x^2) \cdot (x+3) = x^3 + 3x^2. \]

**Division**: \(\frac{f}{g}(x)=\frac{f(a)}{g(a)}\), so for example, \[\frac{b}{c}(x)=\frac{b(x)}{c(x)}=\frac{x+3}{4x}.\]

Note:- The domain of all these combinations is the intersection of the domains of \(f\) and \(g\). That is, both functions must be defined at a point of the combination to be defined. Another requirement for division is that the denominator must be non-zero: in this case \(g(x)\neq0\).

What is \( (f-g)(3) \) if \( f(x) = x^2 + 1 \) and \( g(x) = 2x-2 \)?

Since \( f(3) = 3^2 + 1 = 10 \) and \( g(3) = 2(3) - 2 = 4 \), we can see that

\[ (f-g)(3) = f(3) - g(3) =10 - 4 = 6. \ _\square \]

What is \( (f+g)(4) \) if \( f(x) = 5x^{2} + 4 \) and \( g(x) = 3x+6 \)?

Since \( f(4) = 5\big(4^{2}\big)+4 = 84 \) and \( g(4) = 3(4) +6 = 18\), we can see that

\[ (f+g)(4) = f(4) + g(3) =84 +18 = 102. \ _\square \]

Given \[\begin{align} f(x)&=4{ x }^{ 3 }-5\\ g(x)&=6{ x }^{ 2 }+9\\ h(x)&=(f\cdot g)(x), \end{align}\] what is the value \(h(3)?\)

We know that

\[ h(3)=(f\cdot g)(3)=f(3)\cdot g(3).\]

Since \(f(3)=4(3)^{3}-5=103\) and \(g(3)=6(3)^{2}+9=63,\)

\[\begin{align} h(3)&=f(3)\cdot g(3)\\ &=103\times 63\\ &=6489.\ _\square \end{align}\]

Given \[\begin{align} f(x)&={ x }^{ 2 }+1\\ g(x)&=6x\\ h(x)&=3x+4, \end{align}\] what is the value of \( \big((f\cdot h)-g\big)(3)?\)

The expression can be rewritten as

\[\begin{align} \big((f\cdot h)-g\big)(3)&=(f\cdot h)(3)-g(3)\\ &=f(3)\cdot h(3)-g(3). \end{align}\]

Since \(f(3)=3^{2}+1=10\), \(g(3)=6(3)=18\), and \(h(3)=3(3)+4=13,\) we have

\[f(3)\cdot h(3)-g(3)=10\cdot 13-18=112.\ _\square\]

**Cite as:**Function Arithmetic.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/functions-arithmetic/