When we work with functions, we use a very specific notation to ask for the function's value given a certain input for the independent variable. Additionally, we will be looking at two new scenarios: adding/subtracting two polynomial functions and multiplying/dividing two polynomial functions.

Test Objectives
• Demonstrate an understanding of function composition
• Demonstrate the ability to add two or more functions
• Demonstrate the ability to subtract functions
• Demonstrate the ability to multiply functions
• Demonstrate the ability to divide functions
Operations on Functions Practice Test:

#1:

Instructions: find (f + g)(x) and state the domain.

$$a)\hspace{.2em}$$$$f(x)=x^2 + 5x - 1$$$$g(x)=-x - 3$$

#2:

Instructions: find (fg)(x) and state the domain.

$$a)\hspace{.2em}$$$$f(x)=-4x - 1$$$$g(x)=2x^2 - x$$

#3:

Instructions: find (4f - 5g)(x) and state the domain.

$$a)\hspace{.2em}$$ $$f(x)=9x - 1$$ $$g(x)=5x^2 - x - 2$$

#4:

Instructions: find (fg)(x) and state the domain, find (fg)(5), and (fg)(-1).

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{7x + 1}$$ $$g(x)=3x + 5$$

#5:

Instructions: find (f/g)(x) and state the domain, find (f/g)(5), and (f/g)(2).

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x - 1}$$ $$g(x)=\sqrt{5 - x^2}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ $$(f + g)(x)=x^2 + 4x - 4$$ $$Domain: (-\infty, \infty)$$

#2:

Solutions:

$$a)\hspace{.2em}$$ $$(fg)(x)=-8x^3 + 2x^2 + x$$ $$Domain: (-\infty, \infty)$$

#3:

Solutions:

$$a)\hspace{.2em}$$ $$-25x^2 + 41x + 6$$ $$Domain: (-\infty, \infty)$$

#4:

Solutions:

$$a)\hspace{.2em}$$ $$(fg)(x)=3x\sqrt{7x + 1}+ 5\sqrt{7x + 1}$$ $$Domain: \left[-\frac{1}{7}, \infty\right)$$ $$(fg)(5)=120$$ $$(fg)(-1) \hspace{.2em}is \hspace{.2em}undefined$$

#5:

Solutions:

$$a)\hspace{.2em}$$ $$\left(\frac{f}{g}\right)(x)=\frac{\sqrt{x - 1}}{\sqrt{5 - x^2}}$$ $$Domain: [1, \sqrt{5})$$ $$\left(\frac{f}{g}\right)(5) \hspace{.2em}is \hspace{.2em}undefined$$ $$\left(\frac{f}{g}\right)(2)=1$$