Questions: What is the graph of the function? f(x)=-1/(x+2)+3

Solution Steps

The given function is \( f(x) = \frac{-1}{x+2} + 3 \). This is a rational function with a vertical shift and a horizontal shift.

• Vertical Asymptote: Set the denominator equal to zero: \( x + 2 = 0 \) which gives \( x = -2 \).
• Horizontal Asymptote: As \( x \) approaches infinity, the term \( \frac{-1}{x+2} \) approaches 0, so the horizontal asymptote is \( y = 3 \).

• As \( x \) approaches \( -2 \) from the left, \( f(x) \) approaches \( -\infty \).
• As \( x \) approaches \( -2 \) from the right, \( f(x) \) approaches \( \infty \).
• As \( x \) approaches \( \infty \) or \( -\infty \), \( f(x) \) approaches 3.

Final Answer