Chapter 3, Problem 1MCCP

In Exercises 1-4, graph the given quadratic function. Give each function's domain and range.

$f(x)={(x-3)}^2-4$

To determine

To graph: The function $f\left(x\right)={\left(x-3\right)}^2-4$

Explanation of Solution

Given: $f\left(x\right)={\left(x-3\right)}^2-4$

Graph:

Graph of the function $f\left(x\right)={\left(x-3\right)}^2-4$ is shown below:

Interpretation:

The graph is a parabola

To determine

To Calculate: The domain and range of the provided function $f\left(x\right)={\left(x-3\right)}^2-4$

Answer to Problem 1MCCP

Solution: The domain and range of the function are $\underset{\_}{\left(-\infty ,\infty \right)}$ and $\underset{\_}{\left[-4,\infty \right)}$ respectively

Explanation of Solution

Given: $f\left(x\right)={\left(x-3\right)}^2-4$

Formula used:

Since $f\left(x\right)$ be a polynomial. So, it satisfies for every value of $x$ on the real axis.

Calculation:

Since $f\left(x\right)$ be a polynomial. So, it satisfies for every value of $x$ on the real axis.

Therefore, the domain of $f\left(x\right)={\left(x-3\right)}^2-4$ is $\left(-\infty ,\infty \right)$.

Since,

$\begin{array}{l}{\left(x-3\right)}^2\ge 0\\ \Rightarrow {\left(x-3\right)}^2-4\ge -4\\ \Rightarrow f\left(x\right)\ge -4\\ \Rightarrow f\left(x\right)\in \left[-4,\infty \right)\end{array}$

So, the range of the function is $\left[-4,\infty \right)$.