Questions: What is the graph of the function f(x)=(x^2+3x-4)/(x+4)?

What is the graph of the function f(x)=(x^2+3x-4)/(x+4)?
Transcript text: What is the graph of the function $f(x)=\frac{x^{2}+3 x-4}{x+4}$ ?
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Solution

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Solution Steps

Step 1: Simplify the Function

First, simplify the given function \( f(x) = \frac{x^2 + 3x - 4}{x + 4} \).

Factor the numerator: \[ x^2 + 3x - 4 = (x + 4)(x - 1) \]

So, the function becomes: \[ f(x) = \frac{(x + 4)(x - 1)}{x + 4} \]

Step 2: Cancel Common Factors

Cancel the common factor \( x + 4 \) in the numerator and the denominator: \[ f(x) = x - 1 \]

However, note that \( x \neq -4 \) because the original function has a restriction where the denominator cannot be zero.

Step 3: Identify the Graph

The simplified function \( f(x) = x - 1 \) is a linear function with a slope of 1 and a y-intercept of -1. However, there is a hole in the graph at \( x = -4 \).

Final Answer

The correct graph is the one that represents the line \( y = x - 1 \) with a hole at \( x = -4 \). This corresponds to the first graph.

So, the answer is: \[ \boxed{1} \]

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