Question

sketch the graph of the following function, and state its domain and range. f(x) = -1 + log (x + 6)

Explanation:

Step1: Analyze the parent function

The parent function is \( y = \log(x) \), which has a vertical asymptote at \( x = 0 \), domain \( (0, \infty) \), and range \( (-\infty, \infty) \).

Step2: Analyze the horizontal shift

For \( f(x)=-1+\log(x + 6) \), the argument of the log is \( x + 6 \). To find the vertical asymptote, set \( x+6=0\), so \( x=-6 \). This is a horizontal shift of the parent function \( y = \log(x) \) 6 units to the left.

Step3: Analyze the vertical shift

The \( - 1 \) at the beginning is a vertical shift down by 1 unit.

Step4: Determine the domain

The argument of the logarithm must be positive, so \( x + 6>0\), which gives \( x>-6 \). So the domain is \( (-6,\infty) \).

Step5: Determine the range

For any logarithmic function of the form \( y = a\log(bx + c)+d \), the range is \( (-\infty,\infty) \) (since the logarithm function can take any real value as output after shifts and stretches/compressions, and here there are no stretches/compressions, just shifts). So the range of \( f(x)=-1+\log(x + 6) \) is \( (-\infty,\infty) \).

Step6: Sketching the graph

• Start with the parent graph \( y=\log(x) \), which passes through \( (1,0) \), has a vertical asymptote at \( x = 0 \), and increases slowly.
• Shift the graph 6 units to the left: the point \( (1,0) \) on \( y = \log(x) \) becomes \( (1-6,0)=(-5,0) \) on \( y=\log(x + 6) \). The vertical asymptote shifts from \( x = 0 \) to \( x=-6 \).
• Then shift the graph down 1 unit: the point \( (-5,0) \) becomes \( (-5,0 - 1)=(-5,-1) \). The vertical asymptote remains at \( x=-6 \), and the graph still has the same general shape (increasing, since the coefficient of the log is positive 1) but shifted left and down.

Answer:

• Graph Sketching: The graph is a logarithmic curve with a vertical asymptote at \( x=-6 \), passing through \( (-5,-1) \), increasing (since the log has a positive coefficient), shifted 6 units left and 1 unit down from \( y = \log(x) \).
• Domain: \( (-6,\infty) \)
• Range: \( (-\infty,\infty) \)