Tolnaio

2021-08-12

Name the function $f\left(x\right)=\frac{1}{x}$

pierretteA

The function $f\left(x\right)=\frac{1}{x}$ is called a reciprocal function. Its range is the set of positive real numbers excluding zero, since division by zero is not defined and its domain is the set of real numbers.

Eliza Beth13

To solve the function $f\left(x\right)=\frac{1}{x}$, we need to find the value of $x$ for which $f\left(x\right)$ equals a specific value.
Let's say we want to find the value of $x$ when $f\left(x\right)=a$, where $a$ is a constant.
We can start by setting up the equation:
$\frac{1}{x}=a$
To solve for $x$, we can multiply both sides of the equation by $x$:
$1=ax$
Now, we can isolate $x$ by dividing both sides by $a$:
$x=\frac{1}{a}$
Therefore, the solution to the equation $f\left(x\right)=\frac{1}{x}$ when $f\left(x\right)=a$ is $x=\frac{1}{a}$.

The function can be represented as $f\left(x\right)=\frac{1}{x}$. To solve this function, we need to find the values of $x$ for which $f\left(x\right)$ is defined.
Since division by zero is undefined, we know that $x$ cannot be equal to zero, so we have $x\ne 0$.
To find the inverse of the function, we can solve for $x$ in terms of $f\left(x\right)$:
$y=\frac{1}{x}$
Multiply both sides by $x$ to isolate $x$ on one side:
$yx=1$
Divide both sides by $y$ to solve for $x$:
$x=\frac{1}{y}$
Therefore, the inverse function ${f}^{-1}\left(x\right)$ is given by:
${f}^{-1}\left(x\right)=\frac{1}{x}$
In conclusion, the function $f\left(x\right)=\frac{1}{x}$ is defined for all $x$ except $x=0$, and its inverse function is ${f}^{-1}\left(x\right)=\frac{1}{x}$.

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