Question

read each question carefully. show all of your work for full credit.

1. (4 points) simplify the following exponential expression. your answer should contain only positive exponents.

$3x^{5}y^{2} \cdot 2x^{-2}y^{4} \cdot 5xy^{-9}$

1. (4 points) state if the relation below is a function (explain why/why not) then identify the domain and range.

$\\{(4,12), (- 4,3), (0,9), (- 5,10), (3, - 9), (4,2)\\}$
function? ______
explain:
domain:
range:

1. (2 points) is the inverse of the graph below a function? ______

why/why not?

Explanation:

Question 9

Step1: Multiply the coefficients

Multiply the coefficients \(3\), \(2\), and \(5\). So, \(3\times2\times5 = 30\).

Step2: Multiply the \(x\)-terms

For the \(x\)-terms, we use the rule \(a^m\times a^n=a^{m + n}\). So, \(x^{5}\times x^{-2}\times x^{1}=x^{5+( - 2)+1}=x^{4}\) (since \(x = x^{1}\)).

Step3: Multiply the \(y\)-terms

For the \(y\)-terms, use the same rule \(a^m\times a^n=a^{m + n}\). So, \(y^{2}\times y^{4}\times y^{-9}=y^{2 + 4+( - 9)}=y^{-3}\).

Step4: Convert negative exponent to positive

We know that \(a^{-n}=\frac{1}{a^{n}}\), so \(y^{-3}=\frac{1}{y^{3}}\). Then combine with the other terms: \(30x^{4}\times\frac{1}{y^{3}}=\frac{30x^{4}}{y^{3}}\).

• Function?: A relation is a function if each input (x - value) has exactly one output (y - value). In the given relation \(\{(4,12),(-4,3),(0,9),(-5,10),(3,-9),(4,2)\}\), the input \(x = 4\) is mapped to two different outputs \(12\) and \(2\). So, it is not a function.
• Domain: The domain is the set of all \(x\)-values in the ordered pairs. So, the domain is \(\{-5,-4,0,3,4\}\) (we list each \(x\)-value once, even if it repeats in the ordered pairs).
• Range: The range is the set of all \(y\)-values in the ordered pairs. So, the range is \(\{-9,2,3,9,10,12\}\).

The given graph is a parabola opening upwards (since it has a minimum point). The original graph (a parabola) fails the horizontal line test (a horizontal line will intersect the parabola at more than one point). The inverse of a function is a function if and only if the original function passes the horizontal line test (i.e., the original function is one - to - one). Since the original graph (parabola) does not pass the horizontal line test, its inverse will not be a function. To check the inverse, we can think about reflecting the graph over the line \(y=x\). The original parabola (opening upwards) will have an inverse that is a parabola opening to the right, which will fail the vertical line test (since a vertical line will intersect it at more than one point), meaning the inverse is not a function.

Answer:

\(\frac{30x^{4}}{y^{3}}\)

Question 10