Question

How do you find the range from the undefined graph of $f\left( x \right)={{\log }_{10}}\left( {{x}^{2}}+6x+9 \right)$?

Answer 1

Hint: The range of a function is the set of all the values of the dependent variable y which are obtained after substituting all the values of the independent variable x from the domain of the function. Therefore, the range of the given function $f\left( x \right)={{\log }_{10}}\left( {{x}^{2}}+6x+9 \right)$ can be determined from its graph by noting down all the values from the y-axis, which the graph covers.

Complete step by step solution:
The function given in the question is
$\Rightarrow f\left( x \right)={{\log }_{10}}\left( {{x}^{2}}+6x+9 \right)$
We can put $6x=2\left( 3x \right)$ and $9={{3}^{2}}$ in the argument of the above function to get
$\Rightarrow f\left( x \right)={{\log }_{10}}\left( {{x}^{2}}+2\cdot 3\cdot x+{{3}^{2}} \right)$
Now, we know the algebraic identity given by ${{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}$. Using this identity, we can write the argument of the above function as
$\Rightarrow f\left( x \right)={{\log }_{10}}{{\left( x+3 \right)}^{2}}$
Now, we know that the logarithm function is not defined for the argument of zero. Therefore, the point at which the graph of the given function will not be defined is given by
$\begin{align}
  & \Rightarrow {{\left( x+3 \right)}^{2}}=0 \\
 & \Rightarrow x=-3 \\
\end{align}$
Therefore, the graph of the given function will not be defined at $x=-3$. Hence, the graph of the function will look like

From the above graph we can see that the graph of the given function covers the whole y-axis. Therefore, we can say that the range of the function is $\left( -\infty ,\infty \right)$.
Hence, we have determined the range from the undefined graph of the given function.

Note: We can also determine the range without using the graph by just analyzing the given function. Since the argument to the logarithm function is a polynomial, which takes all the real values, the range of the given function must be the same as that of the function $\log x$, that is $\left( -\infty ,\infty \right)$.