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Question Find the values of xx where the function f(x)=1(x−3)(x−4)f(x)=\frac{1}{(x-3)(x-4)} is not defined and discontinuous.

Studdy Solution

STEP 1

Assumptions
1. The function is given as f(x)=1(x−3)(x−4) f(x) = \frac{1}{(x-3)(x-4)} .
2. A function is not defined where its denominator is equal to zero.
3. A function is discontinuous at points where it is not defined.

STEP 2

Identify the points where the denominator of the function is zero.
The denominator of f(x) f(x) is (x−3)(x−4) (x-3)(x-4) .

STEP 3

Set the denominator equal to zero and solve for x x .
(x−3)(x−4)=0(x-3)(x-4) = 0

STEP 4

Apply the zero product property, which states that if a product of two factors is zero, then at least one of the factors must be zero.
x−3=0orx−4=0x-3 = 0 \quad \text{or} \quad x-4 = 0

STEP 5

Solve the first equation for x x .
x−3=0⇒x=3x-3 = 0 \Rightarrow x = 3

STEP 6

Solve the second equation for x x .
x−4=0⇒x=4x-4 = 0 \Rightarrow x = 4

STEP 7

List the values of x x at which the function is not defined.
The function is not defined at x=3 x = 3 and x=4 x = 4 .
x={3,4}x = \{3, 4\}

STEP 8

Determine the points of discontinuity.
Since the function is not defined at x=3 x = 3 and x=4 x = 4 , these are also the points of discontinuity.
x={3,4}x = \{3, 4\}
The function f(x) f(x) is not defined and discontinuous at x=3 x = 3 and x=4 x = 4 .

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