The values of x for which the function

$f\left(x\right)=\frac{x^2-3x-4}{x^2+3x-4}$

is NOT continuous are

Function Continuity Explanation

The question asks us to find the values of $x$ for which the given function, $f\left(x\right)=\frac{x^2-3x-4}{x^2+3x-4}$, is NOT continuous.

Discontinuity in Rational Functions

A rational function is defined as a ratio of two polynomials. For a rational function to be continuous, its denominator must not be equal to zero. If the denominator of a rational function evaluates to zero at any particular value of $x$, the function becomes undefined at that point, leading to a discontinuity.

Therefore, for the function $f\left(x\right)=\frac{x^2-3x-4}{x^2+3x-4}$ to be discontinuous, its denominator must be equal to zero.

Denominator Analysis

The denominator of the function $f(x)$ is the expression in the lower part of the fraction: $x^2+3x-4$.

To find the values of $x$ where the function is not continuous, we must set the denominator equal to zero and solve for $x$:

$$x^2+3x-4=0$$

Solving the Quadratic Equation

We need to solve the quadratic equation $x^2+3x-4=0$. One common method to solve quadratic equations is by factoring. To factor this quadratic, we look for two numbers that satisfy two conditions:

- When multiplied together, they give the constant term, which is $-4$.
- When added together, they give the coefficient of the $x$ term, which is $3$.

The two numbers that fit these conditions are $4$ and $-1$, because:

- $4\times (-1)=-4$
- $4+(-1)=3$

Using these numbers, we can factor the quadratic expression as follows:

$$(x+4)(x-1)=0$$

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$:

- $x+4=0$
- Solving for $x$, we get $x=-4$.

- $x-1=0$
- Solving for $x$, we get $x=1$.

Conclusion on Discontinuity Points

The values of $x$ for which the denominator of the function becomes zero are $x=-4$ and $x=1$. At these specific $x$ values, the function $f(x)$ is undefined, meaning it is NOT continuous at these points.

Therefore, the function $f(x)$ is not continuous at $x=-4$ and $x=1$.

Matching with Options

Let's compare our calculated values with the given options:

- Option 1: 4 and -1
- Option 2: 4 and 1
- Option 3: -4 and 1
- Option 4: -4 and -1

The values $-4$ and $1$ match Option 3.