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How do you use the horizontal line test to determine whether the function $g\left( x \right)=\dfrac{4-x}{6}$ is one to one?

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Last updated date: 05th Mar 2024
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Hint: We first explain the relation of inverse function with its function types. We find if the function is one-one or not. We use horizontal line tests to determine whether the expression $g\left( x \right)$ is a function or not. We also use a simple formula to find the credibility of the function being one-one.

Complete step-by-step solution:
We need to find whether function $g\left( x \right)=\dfrac{4-x}{6}$ is a one-one function or not. If it is then we find the inverse function.
First, we find the characteristics of the function $g\left( x \right)=\dfrac{4-x}{6}$.
The horizontal line test gives that for the function $g\left( x \right)$ if any horizontal line represented as $y=k$ on infinite extension cuts the graph more than once then the graph can’t have an inverse function.
For $y=g\left( x \right)=\dfrac{4-x}{6}$, we take the horizontal line as $y=5$.
We put $y=5$ in $y=\dfrac{4-x}{6}$ to get $x=4-6y=4-6\times 5=-26$. The line cuts the curve at only one point. Therefore, the function is one-one.
For $y=g\left( x \right)=\dfrac{4-x}{6}$, we have an inverse function which gives $x=4-6y$.
Therefore, the inverse function is ${{g}^{-1}}\left( x \right)=4-6x$.

Note: If the horizontal line intersects the graph of a function in all places at exactly one point, then the given function should have an inverse that is also a function. We say this function passes the horizontal line test.
We can also test the one-one by taking two points of $x$ as $a,b$ where $f\left( a \right)=f\left( b \right)$. If the simplification gives $a=b$ then we say the function is one-one.
$g\left( a \right)=\dfrac{4-a}{6}$ and $g\left( b \right)=\dfrac{4-b}{6}$ which gives
$\begin{align}
  & \dfrac{4-a}{6}=\dfrac{4-b}{6} \\
 & \Rightarrow 4-a=4-b \\
 & \Rightarrow a=b \\
\end{align}$
The function $g\left( x \right)=\dfrac{4-x}{6}$ is one-one.

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