Question

Sketch a graph of the given logarithmic function. $$ f(x)=\log _{10}(x+2) $$

Short answer

The graph has a vertical asymptote at \(x = -2\), an x-intercept at \(x = -1\), and passes through \((0, \log_{10}(2))\).

Step-by-step solution

Identify Domain of the Function

Identify values of \(x\) for which \(f(x) = \log_{10}(x+2)\) is defined. The logarithmic function \(\log_{10}(x+2)\) is defined for \(x+2 > 0\). So, solving \(x+2>0\) gives \(x>-2\). Hence, the domain is \(x > -2\).

Determine Vertical Asymptote

For the logarithmic function \(f(x) = \log_{10}(x+2)\), there is a vertical asymptote where the argument of the logarithm is zero. Thus, we find \(x+2=0\), giving \(x=-2\). This is the vertical asymptote.

Find Intercepts

1. **Y-intercept**: Set \(x=0\) in the function: \(f(0) = \log_{10}(0+2) = \log_{10}(2)\). So the y-intercept is at \((0, \log_{10}(2))\). 2. **X-intercept**: Set \(f(x)=0\): \(\log_{10}(x+2) = 0\) implies \(x+2 = 10^0 = 1\), giving \(x = -1\). Hence, the x-intercept is \((-1, 0)\).

Analyze Behavior as x Approaches the Asymptote and Infinity

1. As \(x\) approaches \(-2\) from the right \((x \to -2^+)\), \(\log_{10}(x+2)\) decreases without bound, moving towards \(-\infty\).2. As \(x\) approaches infinity \((x \to \infty)\), \(\log_{10}(x+2)\) increases without bound, moving towards \(\infty\).

Sketch the Graph

Using the information gathered:1. The vertical asymptote at \(x = -2\).2. The x-intercept at \((-1, 0)\).3. The y-intercept at \((0, \log_{10}(2))\) (approximately 0.301).4. The behavior of the function near the asymptote and as \(x\) approaches infinity.Plot these key points and the general shape of the graph, noting the curve passing through intercepts and increasing towards the right.

Key concepts

Domain and Range

The domain of a logarithmic function is crucial because it tells us the set of all possible input values for the function. For the function \( f(x) = \log_{10}(x+2) \), we begin by ensuring the argument of the log, \( x+2 \), is greater than zero. This requirement comes because the logarithm of a non-positive number is undefined in the realm of real numbers. Hence, solving the inequality \( x+2 > 0 \) yields \( x > -2 \). This means that the domain of the function is all real numbers greater than \(-2\).

In terms of range, logarithmic functions are defined for all real output values. Thus, the range of \( f(x) = \log_{10}(x+2) \) is all real numbers. In summary:

• Domain: \( x > -2 \)
• Range: All real numbers \((-\infty, \infty)\)

Vertical Asymptote

A vertical asymptote in the graph of a function represents a line that the graph approaches but never quite reaches. For the function \( f(x) = \log_{10}(x+2) \), a vertical asymptote is found where the argument of the logarithm equals zero. This occurs because at \( x = -2 \), the expression \( \log_{10}(x+2) \) becomes \( \log_{10}(0) \), which is undefined.

Therefore, \( x = -2 \) is the vertical asymptote for this function. As \( x \) approaches \(-2\) from the right \((x \to -2^+)\), the value of \( \log_{10}(x+2) \) decreases towards \(-\infty\). This behavior is crucial to sketch the graph accurately. It informs us that the graph will fall steeply downwards as it gets closer to this vertical line without ever intersecting it on the left side.

Graph Sketching

When sketching the graph of \( f(x) = \log_{10}(x+2) \), it is helpful to start with several key points and behaviors identified in prior steps:

1. **Vertical Asymptote**: This is at \( x = -2 \). Draw a dashed line here to indicate where the graph will approach but not touch.
2. **Intercepts**:

• **X-intercept**: Occurs at \( (-1, 0) \). At this point, the graph crosses the x-axis.
• **Y-intercept**: Occurs at \( (0, \log_{10}(2)) \) or approximately \((0, 0.301)\). Here, the graph crosses the y-axis.

3. **Behavior Near Infinity**: As \( x \to \infty \), \( f(x) = \log_{10}(x+2) \) increases without bound, indicating that the graph rises continuously to the right.
To sketch the graph:

• Note the curve will start from the vertical asymptote rising sharply, going through the x-intercept at \((-1,0)\).
• It will cross the y-axis at the y-intercept, \((0, 0.301)\), gently curving upwards.
• As \( x \) continues to increase, the curve gradually becomes less steep, continuously rising.

These features together create a curve that is typical for logarithmic graphs, showing both where the function is defined and how it behaves at extreme values.