Question

How do you find the domain and range of $f\left( x \right)=3{{x}^{4}}-10$ ?

Answer 1

We have been given a biquadratic equation in x-variable whose domain and range is to be computed. We can view a function, $f$ as something which takes input $x$ and for a given input, it produces an output which we call $f\left( x \right)$. In order to find the domain, we shall find all the values of x and to find the range, we shall find all the values of $f\left( x \right)$.

Complete step by step solution:
Given that, $f\left( x \right)=3{{x}^{4}}-10$
Any biquadratic function or any polynomial in general does not have any restrictions to the value which shall be given as input to the function. The biquadratic functions are defined for all real numbers which means that any real number can be given as input and substituted in place of x to give the respective output of the function.
Now, to find the range, we shall find the vertex of the graph of the biquadratic function.
Since the coefficient of the 4-degree term, ${{x}^{4}}$ is 3 which is greater than zero, thus, the range of the function is from the vertex of the graph of function to infinity.
The vertex of a quadratic function, $a{{x}^{2}}+bx+c$ is given as $c-\dfrac{{{b}^{2}}}{4a}$.
However for a biquadratic equation, we shall analyze the graph and comment that since it is opening in a vertically upward direction, thus the range can be defined for all ${{x}^{4}}\ge 0$.
Putting $x=0$, we get
$f\left( 0 \right)=3{{\left( 0 \right)}^{4}}-10$
$\Rightarrow f\left( 0 \right)=-10$
Hence the range of the function is $\left[ -10,\infty \right)$.
Therefore, for the function, $f\left( x \right)=3{{x}^{4}}-10$ domain is $x\in \mathbb{R}$ and range is $y\in \left[ -10,\infty \right)$.

Note:
The vertex of a quadratic function, $a{{x}^{2}}+bx+c$ is given as $c-\dfrac{{{b}^{2}}}{4a}$. The vertex for an upward opening parabolic function is the lowest point of the graph whereas the vertex for a downward opening parabolic function is the highest point of the graph.