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.
The given graph of $f$ extends to $-\infty $ in the negative x-direction and to $+\infty $ in the positive x-direction. So, the domain of f is as given by:
$\left( -\infty ,\infty \right)$
Hence, the domain of $f$ is $\left( -\infty ,\infty \right)$.
(b)
The output values 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 was plotted. Hence, the range of the function denotes the y-values of the graph.
The graph of $f$ extends to $-4$ in the negative y-direction and to $+\infty $ in the positive y-direction. So, the range of f is $\left[ -4,\infty \right)$.
Hence, the range of $f$ is $\left[ -4,\infty \right)$.
(c)
The x-intercept is defined as the point where the graph cuts the x-axis. In the given graph $f$ cuts the x-axis at 1 and 7, the x-intercepts are $1\text{ and 7}$.
Hence, the x-intercepts of $f$ are $1\text{ and 7}$.
(d)
The y-intercept is defined as the point where the graph cuts the y-axis. As the graph of $f$ cuts the y-axis at 4 only, so the y-intercept is $4$.
Hence, the y-intercept of $f$ is $4$.
(e)
The function is said to be increasing when with the increase in x-values the corresponding y-values of the graph of the function also increases.
As we see from the graph, the value of the provided function increases in the interval $\left( 4,\infty \right)$.
Hence, the interval in which $f$ is increasing is $\left( 4,\infty \right)$.
(f)
The function is said to be decreasing when the y-values of the graph of the function decrease with the increase in x-values.
As we see from the graph, the value of the provided function decreases in the interval $\left( 0,4 \right)$.
(g)
The function is said to be constant when there is no change in y-values of the graph of the function with the increase in x-values.
Thus, the value of the provided function is constant inthe interval $\left( -\infty ,0 \right)$.
(h)
The relative minimum of a function is defined as that value of x at which the y-value will be minimum as compared to all other points.
Thus, the value of the provided function has a relative minimum at $x=4$.
(i)
The relative minimum of a function is defined as that value of x at which the y-value will be minimum as compared to all other points.
Thus, the value of the provided function has minimum value at $x=4$ , that is, the minimum value of $f$ at $x=4$ is $-4$.
(j)
As seen from the graph, the value of y remains constant for negative values of x. So, the required value is given as below:
$f\left( -3 \right)=4$
Hence, the value of $f\left( -3 \right)$ is $4$.
(k)
As observed from the graph, there are two points for which the y-value is $-2$.
So, the corresponding values of x at which the y-value is −2 are as given below:
$f\left( x \right)=-2 \text{ for } x=2 \text{ and } x=6$
Hence, the values of x for which $f\left( x \right)=-2$ are $2 \text{ and } \text{6}$.
(l)
An even function is the function having symmetry with the y-axis and an odd function is the function having symmetry with the origin .
As the given graph has symmetry with none, so the provided function is neither even nor odd.
Hence, the provided function $f$ is neither even nor odd.
