Question

1. choose the best answer:

you can determine if the inverse of a polynomial function is a function by using the ____ line test on the inverse.
options: vertical, horizontal, reflection

1. choose the best answer:

you can determine if the inverse of a polynomial function is a function by using the ____ line test on the polynomial function.
options: reflection, vertical, horizontal

1. choose the best answer:

the graph of the inverse of a polynomial function is the reflection of the graph of the polynomial function over the line ____.
options: ( y = 0 ), ( x = 0 ), ( y = x )

1. determine the inverse of the function by interchanging the variables and solving for ( y ) in terms of ( x ):

( y = 2x + 1 )
options: ( y = 2x - 1 ), ( y = \frac{1}{2}x + \frac{1}{2} ), ( y = 2x + 1 ), ( y = \frac{1}{2}x - \frac{1}{2} )

Explanation:

Step1: Identify function test for inverse

A function must pass the vertical line test (any vertical line intersects the graph at most once). To check if an inverse graph is a function, apply the vertical line test to it.

Step2: Identify function test for original function

To check if a function has an inverse that is also a function, the original function must pass the horizontal line test (any horizontal line intersects the graph at most once), meaning it is one-to-one.

Step3: Identify inverse reflection line

By definition, the graph of a function's inverse is the reflection of the original function's graph over the line $y=x$.

Step4: Swap variables for inverse

Start with $y=2x+3$. Swap $x$ and $y$:
$x=2y+3$

Step5: Solve for $y$

Isolate $2y$: $x-3=2y$
Divide by 2: $y=\frac{x-3}{2}=\frac{1}{2}x - \frac{3}{2}$

Answer:

1. vertical
2. horizontal
3. $y=x$
4. $y=\frac{1}{2}x - \frac{3}{2}$