using integer values from -5 to 5 , find the interval where the smooth curve of the following function will disconnect:
f(×)=4/×-1
clearly, the break occurs where division by zero is required:
x-1 = 0
x = 1
Well, since we can't divide by zero, we need to find the integer value for "x" that will cause the function to disconnect. Let's plug in the values from -5 to 5 one by one and see what happens.
For x = -5:
f(-5) = 4 / (-5) - 1
f(-5) = -4/5 - 1
f(-5) = -4/5 - 5/5
f(-5) = (-4 - 5) / 5
f(-5) = -9 / 5
For x = -4:
f(-4) = 4 / (-4) - 1
f(-4) = -4/4 - 1
f(-4) = -4/4 - 4/4
f(-4) = (-4 - 4) / 4
f(-4) = -8 / 4
f(-4) = -2
For x = -3:
f(-3) = 4 / (-3) - 1
f(-3) = -4/3 - 1
f(-3) = -4/3 - 3/3
f(-3) = (-4 - 3) / 3
f(-3) = -7 / 3
For x = -2:
f(-2) = 4 / (-2) - 1
f(-2) = -4/2 - 1
f(-2) = -4/2 - 2/2
f(-2) = (-4 - 2) / 2
f(-2) = -6 / 2
f(-2) = -3
For x = -1:
f(-1) = 4 / (-1) - 1
f(-1) = -4/1 - 1
f(-1) = -4 - 1
f(-1) = -5
For x = 0:
f(0) = 4 / 0 - 1
Uh-oh! We can't divide by zero! So, the function disconnects at x = 0.
Since we found an interval where the function disconnects, it happens at x = 0.
To determine the interval where the curve of the function f(x) = 4/(x - 1) will disconnect, we will look for any values of x that create a discontinuity in the function.
The given function has a denominator of (x - 1), so in order for the function to be defined, the denominator cannot be equal to zero.
Setting the denominator equal to zero, we get:
x - 1 = 0
Solving for x, we find that x = 1.
Therefore, the function f(x) will disconnect at x = 1.
Now, let's check the values of x from -5 to 5 to see if the function is defined and continuous within this range.
For x = -5: f(-5) = 4/(-5 - 1) = 4/(-6) = -2/3
For x = -4: f(-4) = 4/(-4 - 1) = 4/(-5) = -4/5
For x = -3: f(-3) = 4/(-3 - 1) = 4/(-4) = -1
For x = -2: f(-2) = 4/(-2 - 1) = 4/(-3) ≈ -1.3333
For x = -1: f(-1) = 4/(-1 - 1) = 4/(-2) = -2
For x = 0: f(0) = 4/(0 - 1) = 4/(-1) = -4
For x = 1: f(1) is undefined (disconnected)
For x = 2: f(2) = 4/(2 - 1) = 4/1 = 4
For x = 3: f(3) = 4/(3 - 1) = 4/2 = 2
For x = 4: f(4) = 4/(4 - 1) = 4/3 ≈ 1.3333
For x = 5: f(5) = 4/(5 - 1) = 4/4 = 1
From the given values of x, we can see that the function is defined and continuous for all values except x = 1. Therefore, the function's smooth curve will disconnect at x = 1.
To find the interval where the smooth curve of the function f(x) = 4/(x - 1) disconnects, we need to look for values of x that would make the denominator zero.
In this case, the function has a discontinuity at x = 1 because it would result in a division by zero error. Therefore, the interval where the smooth curve disconnects would be x ≠ 1.
Let's break it down step by step:
1. Determine the values of x that make the denominator zero:
(x - 1) = 0
x = 1
2. Identify the interval where the smooth curve disconnects:
In our case, the smooth curve of the function disconnects at x = 1 because it results in a division by zero error. We need to exclude this value from the interval. So, the interval becomes x ≠ 1.
Therefore, the smooth curve of the function f(x) = 4/(x - 1) disconnects for all values of x except x = 1.