Question

1. a) on what intervals of x is $f(x)$ increasing?

b) on what intervals of x does $f(x)$ have a negative slope?
c) on what intervals of x is $f(x)$ negative?

1. a) on what intervals of x if $f(x)$ constant?

b) on what intervals of x is $f(x)$ positive?
c) on what intervals of x does $f(x)$ have a positive slope?

1. betty says, \if a function is positive then it is increasing.\ do you agree or disagree with this statement? support your answer.

Explanation:

Problem 8

Step1: Identify increasing interval

A function increases when it rises left to right.
From the graph, $f(x)$ rises for $x < -1$.

Step2: Identify negative slope interval

Negative slope means function falls left to right.
From the graph, $f(x)$ falls for $x > -1$.

Step3: Identify negative $f(x)$ interval

$f(x)$ is negative when below the x-axis.
From the graph, $f(x) < 0$ for $x < -3$.

Step1: Identify constant interval

Constant function is horizontal on graph.
From the graph, $f(x)$ is constant for $-3 < x < 3$.

Step2: Identify positive $f(x)$ interval

$f(x)$ is positive when above the x-axis.
From the graph, $f(x) > 0$ for $x > 3$.

Step3: Identify positive slope interval

Positive slope means function rises left to right.
From the graph, $f(x)$ rises for $x < -3$ and $x > 3$.

A function being positive (its output values are above the x-axis) does not guarantee it is increasing (rising from left to right). For example, a decreasing function like $f(x) = -x + 5$ is positive for $x < 5$, but it slopes downward (decreasing) across this entire interval. This contradicts Betty's statement.

Answer:

a) $(-\infty, -1)$
b) $(-1, \infty)$
c) $(-\infty, -3)$

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Problem 9