Answer
See the full explanation below.
Work Step by Step
(a)
The input values to the function are known as the domain of the function. In the graph, usually x-values are the input values to the function, the graph of which is plotted. Hence, the domain of the function denotes the x-values of the graph.
As the graph of $f$ extends to $-\infty $ in the negative x-direction and to $3$ in the positive x-direction, the domain of f is $\left( -\infty ,3 \right]$.
Hence, the domain of $f$ is $\left( -\infty ,3 \right]$.
(b)
The values that are the output of the function are known as the range of the function. In the graph, usually y-values are the output values of the function, the graph of which is plotted. Hence,the range of the function is denoted by the y-values of the graph .
As the graph of $f$ extends to $-\infty $ in the negative y-direction and to $4$ in the positive y-direction, the range of f is $\left( -\infty ,4 \right]$.
Hence, the range of $f$ is $\left( -\infty ,4 \right]$.
(c)
The zeros of the function are the values of x for which the value of the function is zero. So, the zeros of $f$ are $-\text{3 and 3}$.
Hence, the zeros of $f$ are $-\text{3 and 3}$.
(d)
The value of $f\left( 0 \right)$ is the value of function when $x=0$. So, the value is $f\left( 0 \right)=3$.
Hence, the value $f\left( 0 \right)$ is $3$.
(e)
The function is said to be increasing when with the increase in x-values, there is a corresponding increase in y-values of the graph of the function.
Thus, the value of the provided function increases in the interval $\left( -\infty ,1 \right)$.
Hence, the interval in which $f$ is increasing is $\left( -\infty ,1 \right)$.
(f)
The function is said to be decreasing when with the increase in x-values, there is a decrease in y-values of the graph of the function.
Thus, the value of the provided function decreases in the interval $\left( 1,3 \right)$.
Hence, the interval in which $f$ is decreasing is $\left( 1,3 \right)$.
(g)
The given function has values less than or equal to zero, that is, the value of the provided function $f\left( x \right)\le 0$ is for the values of x that lies in the intervals $\left( -\infty \text{,}-\text{3} \right]$.
Hence, the values of x for which $f\left( x \right)\le 0$ lies in the interval $\left( -\infty \text{,}-\text{3} \right]$.
(h)
The relative maximum of a function will be at that value of x at which y-value will be maximum as compared to all other points.
Thus, the value of the provided function has maximum value at $x=1$ and the relative maxima is 4.
Hence, the provided function has relative maxima at x=1 and the maxima is 4.
(i)
The value of the provided function has value 4 at $x=1$.
Hence, the value of x for which $f\left( x \right)=4$ is 1.
(j)
As the value of y is approximately 2 when x is −1, the value of $f\left( -1 \right)$ is positive.
Hence, the value of $f\left( -1 \right)$ is positive.
