Chapter 2.CT, Problem 1CT

Find the domain of each function. Write each answer using interval notation.

$f(x)=\frac{3}{2x-5}$

To determine

To find:

The domain of the given function and write the answer in interval notation.

Answer to Problem 1CT

Solution:

The domain of the given function is $\left(-\infty ,\frac{5}{2}\right)\cup \left(\frac{5}{2},\infty \right)$.

Explanation of Solution

Given:

$f(x)=\frac{3}{2x-5}$

Approach Used:

A function $f$ is a correspondence between a set of input values $x$ and a set of output values $y$, where to each $x-$value there corresponds exactly one $y-$value.

The domain of a function is the set of all real numbers $x$ for which the function is defined.

Thus, to find the domain of a function,

i. Find the set of numbers that are permissible inputs for $x$.

ii. Determine any restrictions on the input values for $x$.

Calculation:

The domain of $f(x)=\frac{3}{2x-5}$ is the set of all possible values of $x$ i.e. set of real numbers.

Restrictions on the input values for $x$ are any x-values for which $f(x)=0$ or undefined, the $f(x)$ is undefined when the denominator of the function is zero as division by zero is undefined. Hence,

$\begin{array}{l}2x-5=0\\ 2x=5\\ x=\frac{5}{2}\end{array}$

So, the domain is set of all real numbers excluding $\frac{5}{2}$, that is $\left(-\infty ,\frac{5}{2}\right)\cup \left(\frac{5}{2},\infty \right)$.

Final Statement:

Hence, the domain of the given function is $\left(-\infty ,\frac{5}{2}\right)\cup \left(\frac{5}{2},\infty \right)$.