The association between the two determines a function. What are the inputs to this function? the buttons or the letters? What are the outputs? What is the domain of this function? What is the range?
2.
Let \(f(x) = \sqrt{2-x}\text{.}\)
Evaluate \(f\) at \(x=7\text{,}\) if possible.
Evaluate \(f\) at \(x=-7\text{,}\) if possible.
What is the domain of \(f\text{?}\)
When is \(f(x) = 0\text{?}\)
3.Evaluating Functions.
Suppose that \(f(x) = \sqrt{16+x^2}\) and \(g(x) = x^3+2x\text{.}\) Evaluate the expression completely. Show your work carefully as a sequence of equal expressions seperated with the equal sign.
Given the defintions of \(f(x) = 1-2x^2\text{,}\)\(g(x) = \sqrt{2+x}\text{.}\) Find each expression and determine the domain. Then simplify the result, if possible.
\(\displaystyle \displaystyle f-h\)
\(\displaystyle \displaystyle fh\)
\(\displaystyle \displaystyle f/h\)
\(\displaystyle \displaystyle f \circ g \)
\(\displaystyle \displaystyle g \circ f \)
\(\displaystyle \displaystyle f \circ f\)
\(\displaystyle \displaystyle h\circ k\)
\(\displaystyle \displaystyle k\circ h\)
\(\displaystyle \displaystyle g\circ k\)
5.
Let \(F(x) = 1/x\) and \(G(x) = \frac{1-x}{2x+3}\) Find each expression and determine the domain. Then simplify the result, if possible.
\(\displaystyle (F\circ G)(x)\)
\(\displaystyle (G\circ F)(x)\)
\(\displaystyle (G\circ G)(x)\)
\(\displaystyle (G\circF)(x)+2\)
\(\displaystyle (G\circF)(x+2)\)
Inverse Functions.
6.
Suppose \(f\) is a one-to-one function satisfying \(f(0) = -1\text{,}\)\(f(1) = −1\text{,}\) and \(f(−1) = 0\text{.}\) Evaluate each expression below.
\(\displaystyle f^{-1}(1)\)
\(\displaystyle f^{-1}(0)\)
\(\displaystyle -f(1)\)
\(\displaystyle [f(0)]^{-1}\)
\(\displaystyle (f^{-1}\circ f)(0)\)
7.
Given that each function is one-to-one, find its inverse function.
\(\displaystyle f(x) = 4-3x\)
\(\displaystyle g(x) = (4-3x)^3\)
\(\displaystyle h(x) = \frac{1}{4-3x}\)
\(\displaystyle h(x) = \frac{2x+1}{4-3x}\)
\(\displaystyle k(x) = \frac{1}{\sqrt[5]{4-3x}}\)
8.
Explain why \(f(x) = \frac{1}{(2-x)^4}\) is not one-to-one.
9.Difference Quotients.
Find and simplify the difference quotient (6.2) for each function.
