Analyze Function Behavior: Multiplication with Opposite Signs

Question

Determine whether the function described below is increasing, decreasing, or constant:

Each number is multiplied by the same number with different signs.

Video Solution

Solution Steps

00:00 Is the function increasing, decreasing, or constant?
00:03 Let's draw the function graph
00:13 Let's substitute X values and find the corresponding Y values
00:26 Now let's substitute negative X values to check the left side of the graph
00:54 We can see that the function increases until the origin and then decreases
00:58 And this is the solution to the question

Step-by-Step Solution

The function is:

f(x)=x×(x) f(x)=x\times(-x)

Let's start by assuming thatx x equals 0:

f(0)=0×0=0 f(0)=0\times0=0

Now let's assume that x x equals 1:

f(1)=1×1=1 f(1)=1\times-1=-1

Now let's assume that x x equals -1:

f(1)=(1)×(1)=1 f(-1)=(-1)\times(-1)=1

Now let's assume that x x equals 2:

f(2)=2×(2)=4 f(2)=2\times(-2)=-4

Now let's assume that x x equals -2:

f(2)=(2)×(2)=4 f(-2)=(-2)\times(-2)=4

Finally, let's plot all of the points on a graph:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111000

We can see that the function both increases and decreases.

Answer

Increasing and decreasing