Question

school future a2 | 2.09 name: ____ homework | features of graphs period: ____ exercise #3: the entire function y = f(x) is shown on the graph. (a) evaluate each of the following. f(-2)= f(8)= f(-8)= f(4)= (b) are there any local maxima or minima of f(x) in the interval -8 ≤ x ≤ 0? if so, list them as ordered pairs. local max: local min: (c) write the domain and range using inequality notation. domain: range: (d) what are the absolute maximum and minimum values of the function, and at which x - values do they occur? absolute max: f(x)= , at x = , at x = (e) list any x - intercepts and y - intercepts below as ordered pairs. x - intercept(s): y - intercept(s): (f) use the graph to find all values of x that make the equation f(x)=3 true.

Explanation:

Step1: Evaluate function values

To find $f(-2)$, $f(8)$, $f(-8)$, $f(4)$, we look at the $y -$ values of the graph at $x=-2$, $x = 8$, $x=-8$ and $x = 4$ respectively.

Step2: Find local maxima and minima in $-8\leq x\leq0$

Local maxima are points where the function changes from increasing to decreasing in the given interval, and local minima are points where the function changes from decreasing to increasing.

Step3: Determine domain and range

The domain is the set of all $x -$ values for which the function is defined, and the range is the set of all $y -$ values.

Step4: Identify absolute maximum and minimum

The absolute maximum is the highest $y -$ value of the function over its entire domain, and the absolute minimum is the lowest $y -$ value.

Step5: Find intercepts

The $x -$ intercepts are the points where $y = 0$ (where the graph crosses the $x -$ axis), and the $y -$ intercept is the point where $x = 0$ (where the graph crosses the $y -$ axis).

Step6: Solve $f(x)=3$

We find the $x -$ values on the graph where the $y -$ value is 3.

Answer:

(a) Without the actual graph, we can't give numerical values. For example, to find $f(-2)$, we locate $x=-2$ on the $x -$ axis and then find the corresponding $y -$ value on the graph.
(b) Without the graph, we can't list them. But if there is a local max at $x=a$ with $y = b$ in the interval $-8\leq x\leq0$, we list it as $(a,b)$. Similarly for local min.
(c) Without seeing the graph: Domain: If the left - most $x$ value is $x_1$ and the right - most is $x_2$, the domain is $x_1\leq x\leq x_2$. Range: If the lowest $y$ value is $y_1$ and the highest is $y_2$, the range is $y_1\leq y\leq y_2$.
(d) Without the graph, we can't give values. If the absolute max is $y = M$ at $x = m$, we write $f(x)=M$, at $x = m$. Similarly for the absolute min.
(e) $x -$ intercepts: Points $(x_0,0)$ where the graph crosses the $x -$ axis. $y -$ intercept: Point $(0,y_0)$ where the graph crosses the $y -$ axis.
(f) Without the graph, we can't list the $x$ values. We would find the points on the graph where $y = 3$ and list the corresponding $x$ values.