Combinations of Functions Solutions

• When functions are combined, it creates a    new function   .

• Functions can be combined through    addition   ,    subtraction   ,    multiplication   , and    division   .

sum	$\left(f+g\right)\left(x\right)$	$=f\left(x\right)+g\left(x\right)$
difference	$\left(f–g\right)\left(x\right)$	$=f\left(x\right)–g\left(x\right)$
product	$\left(fg\right)\left(x\right)$	$=f\left(x\right)·g\left(x\right)$
quotient	$\left(\frac{f}{g}\right)\left(x\right)$	$=\frac{f\left(x\right)}{g\left(x\right)}, g\left(x\right)\ne 0$

• The    domain    of the combination needs to include the restrictions of    all    functions being combined.

• Pay special attention to    even-degree    and    square root    functions because these have more restrictions on the domains.
• 
The commutative property $(a+b=b+a, ab=ba)$ does not hold true for    subtraction or division   .

• Therefore, make sure to    correctly order    the functions, particularly when determining the difference or quotient.

Example 3

Find $\mathbf{(}\mathit{f}\mathbf{+}\mathit{g}\mathbf{)}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ and $\mathbf{(}\mathit{f}\mathbf{–}\mathit{g}\mathbf{)}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ when$\mathbf{ }\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{–}\mathit{x}\mathbf{–}\mathbf{1}$ and $\mathit{g}\left(\mathit{x}\right)\mathbf{=}\mathit{x}\mathbf{+}\mathbf{6}$. Determine the domain for each function.

Example 4

Find $\mathbf{\left(}\mathit{g}\mathit{h}\mathbf{\right)}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ and $\left(\frac{\mathit{h}}{\mathit{g}}\right)\left(\mathit{x}\right)$ when $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1}$ and $\mathit{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathit{x}\mathbf{–}\mathbf{2}}$. Determine the domain for each function.

$$\begin{gathered} \left(gh\right)\left(x\right)=g\left(x\right)·h\left(x\right) \\ \left(gh\right)\left(x\right)=\left(2x+1\right)\left(\frac{1}{x–2}\right) \\ \left(gh\right)\left(x\right)=\frac{2x+1}{x–2}, x\ne 2 \end{gathered}$$

Note

Check out the More to Explore on combining functions graphically.