Problem 87
Question
For Exercises \(83-90\), the domains were determined in Exercises \(73-80 .\) Use a graph to (a) find the range, (b) give the largest open interval over which the function is increasing, (c) give the largest open interval over which the function is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt[3]{8 x-24}$$
Step-by-Step Solution
Verified Answer
Range is all real numbers; increasing interval: \((-\infty, \infty)\); no decreasing interval; solution for \(f(x)=0\) is \(x=3\).
1Step 1: Graph the Function
To analyze the behavior of the function, first, we'll graph the function \( f(x) = \sqrt[3]{8x - 24} \). This function is a cubic root, which usually has an S-shaped curve. Use a graphing tool or app to input the function and visualize the graph.
2Step 2: Determine the Range
The range of \( f(x) = \sqrt[3]{8x-24} \) can be seen from the graph. Since cube root functions can take any real number output, the range of \( f(x) \) is all real numbers, \( \mathbb{R} \).
3Step 3: Identify Intervals of Increase and Decrease
From observing the graph, identify where the function is going upwards (increasing) or downwards (decreasing). For cubic root functions, they typically increase over their entire domain. Therefore, the largest open interval where \( f(x) \) is increasing is \(( -\infty, \infty)\). There is no interval where the function is decreasing.
4Step 4: Solve \( f(x) = 0 \) by Observing the Graph
Solving \( f(x) = 0 \) means finding the x-values where the graph intersects the x-axis. From the function \( f(x) = \sqrt[3]{8x - 24} \), set \( 8x - 24 = 0 \) and solve for \( x \). This gives \( x = 3 \). You should verify this by observing the graph, ensuring it crosses the x-axis at \( x = 3 \).
Key Concepts
Cubic Root FunctionIncreasing and Decreasing IntervalsSolving Equations GraphicallyGraphing Functions
Cubic Root Function
The cubic root function is a fascinating type of function where we find the cube root of an expression. Unlike the square root, the cubic root can handle both positive and negative numbers. This is because when you cube a number, whether negative or positive, the result is always a real number.
For the function \[ f(x) = \sqrt[3]{8x - 24} \], you are essentially looking for a number, which, when cubed, gives you the expression \(8x - 24\). This function graphs with an S-shaped curve. This curve shows that the function passes through zero and moves indefinitely in both the positive and negative directions on the y-axis.
A cubic root function like this is continuous and smooth without any breaks, defined for all real numbers on the x-axis.
For the function \[ f(x) = \sqrt[3]{8x - 24} \], you are essentially looking for a number, which, when cubed, gives you the expression \(8x - 24\). This function graphs with an S-shaped curve. This curve shows that the function passes through zero and moves indefinitely in both the positive and negative directions on the y-axis.
A cubic root function like this is continuous and smooth without any breaks, defined for all real numbers on the x-axis.
Increasing and Decreasing Intervals
Understanding where a function is increasing or decreasing on its graph is crucial. For a cubic root function such as \[ f(x) = \sqrt[3]{8x - 24} \], we see a consistent upward trend throughout its domain. This means it is always increasing.
### Increasing IntervalsIn our function, from any point to its right on the x-axis, the graph is going upwards. This denotes that it is increasing across the interval \((-\infty, \infty)\). This entire x-axis interval is where the function climbs without a single descent in the graph's journey.
### Decreasing IntervalsInterestingly, for this specific cubic root function, no part of the graph goes downward from left to right. Therefore, there are no intervals where this function decreases. This is typical of many cubic root functions, which predominantly exhibit a purely increasing pattern.
### Increasing IntervalsIn our function, from any point to its right on the x-axis, the graph is going upwards. This denotes that it is increasing across the interval \((-\infty, \infty)\). This entire x-axis interval is where the function climbs without a single descent in the graph's journey.
### Decreasing IntervalsInterestingly, for this specific cubic root function, no part of the graph goes downward from left to right. Therefore, there are no intervals where this function decreases. This is typical of many cubic root functions, which predominantly exhibit a purely increasing pattern.
Solving Equations Graphically
When faced with an equation like \[ f(x) = 0 \], it’s useful to rely on the graph to find solutions. For the function \[ \sqrt[3]{8x - 24} \], solving for when the function equals zero involves identifying where the graph intercepts the x-axis.
Here's how you do that: Set the expression inside the cubic root to zero, leading to \(8x - 24 = 0\). Solving this gives us \[ x = 3 \]. This means the function crosses the x-axis at the point \(x = 3\). Verifying this on the graph can confirm your algebraic solution. Seeing the graph intersect at \(x = 3\) provides concrete visual evidence that you've found the correct solution.
Here's how you do that: Set the expression inside the cubic root to zero, leading to \(8x - 24 = 0\). Solving this gives us \[ x = 3 \]. This means the function crosses the x-axis at the point \(x = 3\). Verifying this on the graph can confirm your algebraic solution. Seeing the graph intersect at \(x = 3\) provides concrete visual evidence that you've found the correct solution.
Graphing Functions
Graphing a function is like plotting its life story on an x and y-axis. For \[ f(x) = \sqrt[3]{8x - 24} \], the graph helps illustrate key concepts like the function's behavior, range, and zero points.
To plot this graph, employ a graphing calculator or software by entering the equation directly. The resulting curve should resemble an elongated "S" across the y-axis.
### Important Features of Graphs- **Domain and Range:** Understanding that the cubic root function operates over all real numbers means you'll see both very high and very low points. - **Intercepts:** Look for where the graph crosses important axes—here, finding that x-intercept proves critical for solving our zero equations.- **Intervals:** Examine closely which parts of the graph are moving up or down (though as we've seen, here it is only upwards).
Effective graphing provides both a tool and a visualization, helping us not only to solve equations but to see a function's broader characteristics.
To plot this graph, employ a graphing calculator or software by entering the equation directly. The resulting curve should resemble an elongated "S" across the y-axis.
### Important Features of Graphs- **Domain and Range:** Understanding that the cubic root function operates over all real numbers means you'll see both very high and very low points. - **Intercepts:** Look for where the graph crosses important axes—here, finding that x-intercept proves critical for solving our zero equations.- **Intervals:** Examine closely which parts of the graph are moving up or down (though as we've seen, here it is only upwards).
Effective graphing provides both a tool and a visualization, helping us not only to solve equations but to see a function's broader characteristics.
Other exercises in this chapter
Problem 86
Solve each problem. If \(y\) varies directly with \(x\) and inversely with \(m^{2}\) and \(r^{2},\) and \(y=\frac{5}{3}\) when \(x=1, m=2,\) and \(r=3,\) find \
View solution Problem 87
Solve each problem. For \(k>0,\) if \(y\) varies directly with \(x,\) when \(x\) increases, \(y\) ______\(,\) and when \(x\) decreases, \(y\) ______.
View solution Problem 88
Solve each problem. For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _______ and when \(x\) decreases, \(y\) _______.
View solution Problem 88
For Exercises \(83-90\), the domains were determined in Exercises \(73-80 .\) Use a graph to (a) find the range, (b) give the largest open interval over which t
View solution