Problem 17
Question
Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\)
Step-by-Step Solution
Verified Answer
The average rate of change of the function \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\) is \(\frac{1}{5}\)
1Step 1: Substitute values into the function
Substitute \(x_{1}\) and \(x_{2}\) into the function \(f(x)\). This gives you \(f(x_{1})=f(4)=\sqrt{4}=2\) and \(f(x_{2})=f(9)=\sqrt{9}=3\)
2Step 2: Substitute into the average rate of change formula
Substitute the values obtained in Step 1 as well as \(x_{1}\) and \(x_{2}\) into the formula for the average rate of change. The formula of average rate of change is given by \(\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\) . Hence, the average rate of change is \(\frac{3-2}{9-4}=\frac{1}{5}\)
Key Concepts
Functions in AlgebraSquare Root FunctionsRate of Change Calculation
Functions in Algebra
In algebra, functions are used to describe the relationship between two sets of numbers or variables. The function takes one or more inputs (often represented as 'x') and produces a single output (often represented as 'f(x)'). A key feature of functions is that each input is related to exactly one output. An easy way to visualize a function is by imagining it as a machine that takes an input, does something to it, and then gives an output.
For example, with the square root function, which is a type of algebraic function, every non-negative input has a non-negative square root. When working with functions, we often look at graphs on a coordinate plane, where the horizontal axis (x-axis) represents the inputs and the vertical axis (y-axis) is for the outputs. By plotting points on the graph where the input and corresponding output meet, one can see the shape of the function and understand how the output changes as the input changes.
For example, with the square root function, which is a type of algebraic function, every non-negative input has a non-negative square root. When working with functions, we often look at graphs on a coordinate plane, where the horizontal axis (x-axis) represents the inputs and the vertical axis (y-axis) is for the outputs. By plotting points on the graph where the input and corresponding output meet, one can see the shape of the function and understand how the output changes as the input changes.
Square Root Functions
A square root function is a specialized type of function in algebra defined as f(x) = \(\sqrt{x}\). It is represented graphically as a curve that starts at the origin (0,0) if the domain is restricted to non-negative numbers. The function outputs the principal square root of the input value. This function is unique because it involves a radical expression, specifically the square root.
To better grasp square root functions, it's essential to remember that the square root of a number is a value that, when multiplied by itself, gives the original number. For example, since 2 * 2 = 4, the square root of 4 is 2, written as \(\sqrt{4}=2\). This concept becomes vital when you're finding the average rate of change in a function over a specific interval because you'll be dealing with the actual output values of the function.
To better grasp square root functions, it's essential to remember that the square root of a number is a value that, when multiplied by itself, gives the original number. For example, since 2 * 2 = 4, the square root of 4 is 2, written as \(\sqrt{4}=2\). This concept becomes vital when you're finding the average rate of change in a function over a specific interval because you'll be dealing with the actual output values of the function.
Rate of Change Calculation
The rate of change in mathematics signifies how one quantity changes in relation to another. In the context of functions, the average rate of change is similar to the concept of slope in geometry—it's the change in the function's output values divided by the change in the input values over a particular interval.
To calculate the average rate of change, you take the difference between the function values at the end and the beginning of the interval and divide by the difference in the input values over the same interval. This is expressed mathematically as \(\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\). In our exercise, the average rate of change of the square root function \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\) is calculated by finding the function values at these points and then applying the formula, yielding \(\frac{3-2}{9-4}=\frac{1}{5}\). This shows that, on average, the function's output increases by 0.2 for each unit increase in x within the interval from 4 to 9.
To calculate the average rate of change, you take the difference between the function values at the end and the beginning of the interval and divide by the difference in the input values over the same interval. This is expressed mathematically as \(\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\). In our exercise, the average rate of change of the square root function \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\) is calculated by finding the function values at these points and then applying the formula, yielding \(\frac{3-2}{9-4}=\frac{1}{5}\). This shows that, on average, the function's output increases by 0.2 for each unit increase in x within the interval from 4 to 9.
Other exercises in this chapter
Problem 17
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Determine whether each equation defines \(y\) as a function of \(x .\) $$ x=y^{2} $$
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Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-4,\) passing through \((-4,0)\)
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Determine whether each function is even, odd, or neither. $$ \begin{aligned} &f(x)=x^{3}+x\\\ &g(x)=x^{2}+x\\\ &h(x)=x^{2}-x^{4}\\\ &f(x)=x^{2}-x^{4}+1\\\ &f(x)
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