Question

Use the guidelines of this section to make a complete graph of \(f\). $$f(x)=x \sqrt{x+3}$$

Short answer

Answer: As x approaches -3 from the right, f(x) approaches 0. As x approaches positive infinity, f(x) increases without bound. The intercepts of the graph are x=-3, x=0, and y=0.

Step-by-step solution

Find the domain of the function

The domain of the function determines the set of x-values for which the function is defined. Since the function has a square root, we need to make sure the value under the square root is non-negative. In other words, \(x+3 \ge 0\). To find the domain, solve for x: $$x+3 \ge 0$$ $$x \ge -3$$ Therefore, the domain of the function is \(x \ge -3\).

Find the intercepts

Intercepts are points where the function touches or crosses the x-axis or y-axis. - To find the x-intercept, set \(f(x) = 0\) and solve for x: $$0 = x \sqrt{x+3}$$ x-intercept occurs when \(x=0\) or \(x=-3\). - To find the y-intercept, set \(x = 0\) and solve for f(x): $$f(0) = (0) \sqrt{0+3} = 0$$ y-intercept occurs at y=0. The intercepts are \(x=-3\), \(x=0\), and \(y=0\).

Find the first derivative and critical points

Find the first derivative of the function to determine where \(f'(x) = 0\) or is undefined. These points are the critical points of the function. $$f'(x) = \frac{d}{dx} (x \sqrt{x+3})$$ Using the product rule \(\frac{d}{dx}(uv) = u'v + uv'\), where \(u=x,\) and \(v=\sqrt{x+3}\): $$u' = 1$$ To find \(v'\), we rewrite \(v\) as \((x+3)^{\frac{1}{2}}\) and apply the chain rule: $$v' = \frac{1}{2}(x+3)^{-\frac{1}{2}} \cdot (1) = \frac{1}{2\sqrt{x+3}}$$ Now, calculate \(f'(x)\): $$f'(x) = u'v + uv' = (1) \cdot (\sqrt{x+3}) + (x) \cdot \left( \frac{1}{2\sqrt{x+3}} \right) = \sqrt{x+3} + \frac{x}{2\sqrt{x+3}}$$ The critical points occur when \(f'(x) = 0\) or does not exist. \(f'(x)\) does not exist when the denominator is 0 , which occurs at x = -3. However, -3 is an endpoint of the domain, so it's not considered a critical point. Set \(f'(x) = 0\) and solve for x: $$0 = \sqrt{x+3} + \frac{x}{2\sqrt{x+3}}$$ No further simplification and factoring are possible here to find the critical points analytically. You can use numerical methods or graphic calculators to find that there is no critical point for this function.

Analyze the behavior of the function as it approaches infinity

As x approaches positive infinity, both x and the square root x+3 grow without bound. This means f(x) will also increase without bound. As x approaches -3 from the right (since the domain is x≥-3), the square root x+3 approaches 0, and f(x) approaches 0 as well since x=-3.

Plot the graph using the obtained information

Using the information obtained in the previous steps, we can now sketch the graph of the function \(f(x) = x \sqrt{x+3}\): 1. The domain is x ≥ -3. 2. The intercepts are x=-3, x=0, and y=0. 3. There are no critical points in the domain. 4. When x approaches positive infinity, f(x) increases without bound. 5. When x approaches -3 from the right, f(x) approaches 0. With these data points and understanding of the function's behavior, you can now plot the graph of \(f(x) = x \sqrt{x+3}\).

Key concepts

Domain of a function

The domain of a function refers to the complete set of possible values of the independent variable, usually denoted as \( x \), which makes the function define and work properly. For the function \( f(x) = x \sqrt{x+3} \), it's crucial that the expression under the square root stays non-negative. Usually, we require any expression inside a square root to be zero or positive. Thus, we solve \( x+3 \ge 0 \) to find the domain.
\[ x+3 \ge 0 \]
Simplifying this gives us:

• Subtract 3 from both sides to get \( x \ge -3 \).

This means \( x \) can be any number that is greater than or equal to -3. Thus, the domain of the function is \( x \ge -3 \). In simpler terms, the function works anywhere from -3 onwards to infinity.

Intercepts in graphing

Intercepts are points where the graph crosses the axes. Finding intercepts is an important skill in graphing because they provide anchor points for plotting the function.

To find the **x-intercept**, set the function equal to zero and solve for \( x \). For \( f(x) = x \sqrt{x+3} \):

• Set \( 0 = x \sqrt{x+3} \).
• Solve to find \( x = 0 \) or \( x = -3 \) after factoring out the terms.

The x-intercepts are at the points \( x = -3 \) and \( x = 0 \).

For the **y-intercept**, let \( x = 0 \) and solve for \( f(0) \):

• The function becomes \( f(0) = 0 \sqrt{0+3} = 0 \).

The y-intercept occurs at \( y = 0 \). This means the graph passes through the origin (0,0). These intercepts help in sketching the curve accurately.

Derivative calculus

Derivative calculus involves finding the rate at which a function is changing at any given point. For \( f(x) = x \sqrt{x+3} \), we use the derivative to find critical points that might indicate local maxima or minima.

To find the derivative \( f'(x) \), apply the product rule. If you have a function that is the product of two functions, \( uv \), its derivative is \( u'v + uv' \). Here, let \( u = x \) and \( v = \sqrt{x+3} \).

• Differentiate \( u=x \) to get \( u' = 1 \).
• Express \( v \) as \( (x+3)^{\frac{1}{2}} \) and differentiate using the chain rule to obtain \( v' = \frac{1}{2\sqrt{x+3}} \).

Combining these through the product rule gives:
\[ f'(x) = 1 \cdot \sqrt{x+3} + x \cdot \frac{1}{2\sqrt{x+3}} \]
This derivative helps us understand the behavior of the function and potential points of interest like maxima, minima, or inflections, which are root indicators for understanding how a function behaves.

Analyzing asymptotic behavior

Asymptotic behavior analysis is about understanding the behavior of a function as the input approaches certain critical values, such as infinity or points near the edge of its domain.

**As \( x \to \infty \):**
For our function, as \( x \) becomes larger and larger, both \( x \) and \( \sqrt{x+3} \) tend to increase without bound. Consequently, the product \( f(x) = x \sqrt{x+3} \) also increases indefinitely, indicating:

• The function has no horizontal asymptote.

**As \( x \to -3^+ \) (from the right):**
Approaching \( x = -3 \) from the right within its domain, \( \sqrt{x+3} \) tends toward zero because \( x+3 \) approaches zero. The function roughly behaves near zero at this point:

• This is not an asymptote in the traditional sense but tells us that \( f(x) \) falls to zero as \( x \) nears -3 from the right.

Understanding asymptotic behavior helps in predicting how the function stretches or flattens across its domain."}]}]} This appropriately explores the function's end behavior within its domain. By learning these concepts, we gain valuable insights into function characteristics and graphing them accurately.