Questions: If a function f is increasing on the intervals [-1,5] and [11,21] and decreasing on the interval [5,11], determine the interval (s) on which g(x)=-3 f(2 x-3) is increasing.

If a function f is increasing on the intervals [-1,5] and [11,21] and decreasing on the interval [5,11], determine the interval (s) on which g(x)=-3 f(2 x-3) is increasing.

Transcript text: If a function $f$ is increasing on the intervals $[-1,5]$ and $[11,21]$ and decreasing on the interval $[5,11]$, determine the interval $(s)$ on which $g(x)=-3 f(2 x-3)$ is increasing.

Solution

Solution Steps

To determine the intervals on which the function $ g(x) = -3 f(2x - 3) $ is increasing, we need to analyze the transformations applied to the function $ f(x) $. The function $ g(x) $ involves a horizontal scaling by a factor of $ \frac{1}{2} $, a horizontal shift to the right by $ \frac{3}{2} $, and a vertical reflection and scaling by a factor of $-3$. The horizontal scaling and shift affect the intervals of increase and decrease, while the vertical reflection reverses the direction of increase and decrease.

Horizontal Scaling and Shifting: The transformation $ 2x - 3 $ scales the x-values by $ \frac{1}{2} $ and shifts them to the right by $ \frac{3}{2} $. This means the intervals of increase and decrease for $ f(x) $ will be transformed accordingly.
Vertical Reflection: The multiplication by $-3$ reflects the function vertically, which means intervals where $ f(x) $ is increasing will become intervals where $ g(x) $ is decreasing, and vice versa.

Given that $ f(x) $ is increasing on $[-1, 5]$ and $[11, 21]$, and decreasing on $[5, 11]$, we need to find the corresponding intervals for $ g(x) $.

Step 1: Identify Intervals for $ f(x) $

The function $ f(x) $ is given to be increasing on the intervals:

$ I_1: [-1, 5] $
$ I_2: [11, 21] $

It is decreasing on the interval:

$ D: [5, 11] $

Step 2: Apply Transformations to Find Intervals for $ g(x) $

The function $ g(x) = -3 f(2x - 3) $ involves two transformations:

Horizontal Scaling and Shifting: The transformation $ 2x - 3 $ scales the x-values by $ \frac{1}{2} $ and shifts them to the right by $ \frac{3}{2} $.
Vertical Reflection: The multiplication by $-3$ reflects the function vertically, reversing the intervals of increase and decrease.

Step 3: Transform Increasing Intervals

For the increasing intervals of $ f(x) $:

For $ I_1: [-1, 5] $: \[ \text{New interval} = \left[ \frac{-1 + 3}{2}, \frac{5 + 3}{2} \right] = \left[ 1, 4 \right] \]
For $ I_2: [11, 21] $: \[ \text{New interval} = \left[ \frac{11 + 3}{2}, \frac{21 + 3}{2} \right] = \left[ 7, 12 \right] \]

Thus, after transformation, $ g(x) $ is decreasing on the intervals $ [1, 4] $ and $ [7, 12] $.

Step 4: Transform Decreasing Intervals

For the decreasing interval of $ f(x) $: