About Operations on Functions:
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
#1:
Instructions: find (f + g)(x) and state the domain.
$$a)\hspace{.2em}$$$$ f(x)=x^2 + 5x - 1 $$$$ g(x)=-x - 3$$
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#2:
Instructions: find (fg)(x) and state the domain.
$$a)\hspace{.2em}$$$$ f(x)=-4x - 1 $$$$g(x)=2x^2 - x$$
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#3:
Instructions: find (4f - 5g)(x) and state the domain.
$$a)\hspace{.2em}$$ $$f(x)=9x - 1$$ $$g(x)=5x^2 - x - 2$$
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#4:
Instructions: find (fg)(x) and state the domain, find (fg)(5), and (fg)(-1).
$$a)\hspace{.2em}$$ $$f(x)=\sqrt{7x + 1}$$ $$g(x)=3x + 5$$
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#5:
Instructions: find (f/g)(x) and state the domain, find (f/g)(5), and (f/g)(2).
$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x - 1}$$ $$g(x)=\sqrt{5 - x^2}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$(f + g)(x)=x^2 + 4x - 4$$ $$Domain: (-\infty, \infty)$$
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$(fg)(x)=-8x^3 + 2x^2 + x$$ $$Domain: (-\infty, \infty)$$
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#3:
Solutions:
$$a)\hspace{.2em}$$ $$-25x^2 + 41x + 6$$ $$Domain: (-\infty, \infty)$$
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#4:
Solutions:
$$a)\hspace{.2em}$$ $$(fg)(x)=3x\sqrt{7x + 1}+ 5\sqrt{7x + 1}$$ $$Domain: \left[-\frac{1}{7}, \infty\right)$$ $$(fg)(5)=120$$ $$(fg)(-1) \hspace{.2em}is \hspace{.2em}undefined$$
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#5:
Solutions:
$$a)\hspace{.2em}$$ $$\left(\frac{f}{g}\right)(x)=\frac{\sqrt{x - 1}}{\sqrt{5 - x^2}}$$ $$Domain: [1, \sqrt{5})$$ $$\left(\frac{f}{g}\right)(5) \hspace{.2em}is \hspace{.2em}undefined$$ $$\left(\frac{f}{g}\right)(2)=1$$
