Suppose f:D⟶R*f*:*D*⟶R is a function defined by f(x)=x2−16x+4*f*(*x*)=*x*+4*x*2−16​​, where D⊂R*D*⊂R. Let A*A* be the set of integers which are not in the domain of f*f*, then find the cardinality of the set A

The function given is:

\[
f(x) = \frac{x^2 - 16}{x + 4}
\]

### Step 1: Simplify the function

First, observe that the denominator \(x + 4\) cannot be zero because division by zero is undefined. So, we need to find when the denominator is zero:

\[
x + 4 = 0 \implies x = -4
\]

Thus, \(x = -4\) is not in the domain of the function.

### Step 2: Factorize the numerator

Now, let's factorize the numerator \(x^2 - 16\):

\[
x^2 - 16 = (x - 4)(x + 4)
\]

So, the function can be rewritten as:

\[
f(x) = \frac{(x - 4)(x + 4)}{x + 4}
\]

For \(x \neq -4\), the \(x + 4\) terms cancel out, leaving:

\[
f(x) = x - 4 \quad \text{for} \quad x \neq -4
\]

### Step 3: Identify the domain

The only value of \(x\) for which the function is not defined is \(x = -4\). So, the domain of the function is all real numbers except \(x = -4\):

\[
D = \mathbb{R} \setminus \{-4\}
\]

### Step 4: Find the set \(A\)

The set \(A\) is defined as the set of integers that are not in the domain of \(f\). Since the only point excluded from the domain is \(x = -4\), the set \(A\) is:

\[
A = \{-4\}
\]

### Step 5: Determine the cardinality of \(A\)

The set \(A = \{-4\}\) contains only one element, so the cardinality of \(A\) is:

\[
\boxed{1}
\]

Would you like further details on any step or have more questions?

### Related Questions:
1. What is the range of the function \(f(x) = x - 4\) for \(x \neq -4\)?
2. How do you find the limit of \(f(x)\) as \(x\) approaches -4?
3. Can the function \(f(x) = \frac{x^2 - 16}{x + 4}\) have any vertical or horizontal asymptotes?
4. How does removing \(x = -4\) from the domain affect the continuity of the function?
5. What are the critical points of the simplified function \(f(x) = x - 4\)?

### Tip:
Always check the denominator for values that make the function undefined when simplifying rational expressions.

Suppose f:D⟶R is a function defined by f(x)=x2−16x+4, where D⊂R. Let A be the set of integers which are not in the domain of f, then find the cardinality of the set A.

Explore how to find the domain of the rational function f(x) = (x^2 - 16) / (x + 4), identifying values that make the function undefined. This guide also calculates the cardinality of the set of integers not in the domain, using key concepts in algebra and rational expressions. Suitable for Grades 10-12.

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