Question

section 1.3 more on functions and their graphs 197
where

1. use the graph of f to determine each of the following. where applicable, use interval notation.

a. the domain of f
b. the range of f
c. the zeros of f
d. f(0)
e. intervals on which f is increasing
f. intervals on which f is decreasing
g. intervals on which f is constant
h. values of x for which f(x)>0
i. values of x for which f(x)= - 2
j. is f(4) positive or negative?
k. is f even, odd, or neither?
l. is f(2) a relative maximum?
in exercises 53 - 58, evaluate each piecewise function at the given values of the independent variable.

1. f(x) = {3x + 5 if x<0; 4x + 7 if x≥0}

Explanation:

Step1: Identify domain from graph

Domain is the set of all x - values. From the graph, x ranges from - 5 to 5.
$[-5,5]$

Step2: Identify range from graph

Range is the set of all y - values. The lowest y - value is - 3 and the highest is 2.
$[-3,2]$

Step3: Find zeros of f

Zeros are x - values where y = 0. From the graph, x=-4 and x = 4.
$x=-4,4$

Step4: Find f(0)

When x = 0, y = 1.
$f(0)=1$

Step5: Find increasing intervals

Increasing means as x increases, y increases. The function is increasing on $[-5,-2]$.
$[-5,-2]$

Step6: Find decreasing intervals

Decreasing means as x increases, y decreases. The function is decreasing on $[2,5]$.
$[2,5]$

Step7: Find constant intervals

Constant means y - value does not change with x. The function is constant on $[-2,2]$.
$[-2,2]$

Step8: Find x for f(x)>0

y>0 when - 4 < x < 4.
$(-4,4)$

Step9: Find x for f(x)=-2

y=-2 when x=-5 and x = 5.
$x=-5,5$

Step10: Determine f(4) sign

At x = 4, y = 0, so it is neither positive nor negative.
Neither

Step11: Check if f is even, odd or neither

$f(-x)
eq f(x)$ and $f(-x)
eq - f(x)$, so it is neither.
Neither

Step12: Check if f(2) is relative maximum

f(2) is part of a constant - value section, not a relative maximum.
No

Answer:

a. $[-5,5]$
b. $[-3,2]$
c. $x=-4,4$
d. $f(0)=1$
e. $[-5,-2]$
f. $[2,5]$
g. $[-2,2]$
h. $(-4,4)$
i. $x=-5,5$
j. Neither
k. Neither
l. No