Question

mat100 warm - up
gidd starts this evening
good morning we are working on the warm - up :)

1. is a circle a function? why or why not?
2. find (f^{-1}(x))

(f(x)=7x - 6)

1. find (f^{-1}(x)=\frac{2 - x}{5})

3 write the conditional statements and the symbolic notation.
original : when it is day the sun is out.
conditional:
inverse:
converse :
contrapositive :

Explanation:

1. Is a circle a function? Why or why not?

Step1: Recall function definition

A function is a relation where each input (x - value) has exactly one output (y - value).

Step2: Consider vertical - line test

For a circle given by the equation $(x - a)^2+(y - b)^2=r^2$, if we apply the vertical - line test, a vertical line can intersect the circle at two points. For example, for the unit circle $x^{2}+y^{2}=1$, when $x = 0$, $y=\pm1$.

Step1: Let $y = f(x)$

Set $y = 7x-6$.

Step2: Solve for $x$ in terms of $y$

Add 6 to both sides: $y + 6=7x$. Then divide both sides by 7, we get $x=\frac{y + 6}{7}$.

Step3: Swap $x$ and $y$

The inverse function $f^{-1}(x)=\frac{x + 6}{7}$.

Step1: Let $y = f(x)$

We have $y=\frac{2 - x}{5}$.

Step2: Solve for $x$ in terms of $y$

Multiply both sides by 5: $5y=2 - x$. Then add $x$ to both sides and subtract $5y$ from both sides to get $x = 2-5y$.

Step3: Swap $x$ and $y$

The inverse function $f^{-1}(x)=2 - 5x$.

Answer:

No, a circle is not a function because it fails the vertical - line test. There are x - values that correspond to more than one y - value.

2. Find $f^{-1}(x)$ for $f(x)=7x - 6$