Identify any x-values at which the absolute value function f(x) = 6 |x + 7|, is
not continuous: x = _____
not differentiable: x = _____
The function is continuous everywhere but not differentiable at x = -7.
Plot it and you will see why. The slope abruptly changes at x = -7
Thanks
To determine the x-values at which the absolute value function f(x) = 6 |x + 7| is not continuous, we need to look for any potential discontinuities.
The absolute value function |x + 7| is continuous for all real numbers x. However, we also need to take into consideration any values of x that make the whole function not continuous.
Since the multiplication by 6 does not introduce any new potential discontinuities, we just need to focus on the argument of the absolute value.
The absolute value function becomes discontinuous when the argument, x + 7, equals zero. Therefore, to find the x-values at which f(x) is not continuous, we solve the equation x + 7 = 0.
x + 7 = 0
x = -7
Hence, the absolute value function f(x) = 6 |x + 7| is not continuous at x = -7.
Now, to determine the x-values at which f(x) is not differentiable, we need to look for any potential corners, cusps, or sharp turns in the graph.
The graph of the absolute value function has a sharp turn (a corner) at x = -7, where the function changes direction abruptly.
Therefore, at x = -7, the absolute value function f(x) = 6 |x + 7| is not differentiable.
To identify any x-values at which the absolute value function f(x) = 6 |x + 7| is not continuous, we need to look for any values of x that make the function not defined or cause a jump or discontinuity in the graph.
For the function f(x) = 6 |x + 7|, we know that the absolute value function |x + 7| will be equal to 0 when x + 7 = 0, which means x = -7. Therefore, at x = -7, the function f(x) is not defined, since the absolute value of -7+7 is 0 and multiplying it by 6 gives us 0.
Therefore, x = -7 is the x-value at which the absolute value function is not continuous.
To identify any x-values at which the function f(x) = 6 |x + 7| is not differentiable, we need to check if the derivative of the function exists and is defined at every point.
The derivative of the absolute value function |x + 7| is not defined when the slope changes abruptly or there is a sharp corner in the graph. For the absolute value function, both of these cases occur when the value inside the absolute value expression changes sign.
In this case, the expression inside the absolute value is (x + 7). It will change sign at x = -7, as we discussed earlier. Therefore, at x = -7, the function f(x) is not differentiable.
So, x = -7 is also the x-value at which the absolute value function is not differentiable.
To summarize:
Not continuous: x = -7
Not differentiable: x = -7