Question

\(f(x)=\sqrt{x}, g(x)=\sqrt{x+1}+8\)

Short answer

The domain of function \(f(x) = \sqrt{x}\) is \([0, +\infty)\) and the function \(g(x) = \sqrt{x + 1} + 8\) is \([-1, +\infty)\). Both functions are similar shapes, but \(g(x)\) is shifted one unit to the left and eight units up compared to \(f(x)\). These functions do not intersect, so no solution exists where they intersect.

Step-by-step solution

Determine the Domain of the Functions

The domain of \(f(x) = \sqrt{x}\) is \([0, +\infty)\) because the square root of a real number is defined only for that number and greater than zero. In the \(g(x) = \sqrt{x + 1} + 8\), the domain is \([-1, +\infty)\) because the value inside the square root, \(x+1\), should be greater than or equal to zero.

Plot the Functions

The plot of \(f(x)\) is a curve starting from the origin and growing. \(g(x)\) is similar as \(f(x)\), but it is translated one unit to the left and eight units up.

Find the Intersection of Functions

To find the intersections of two functions, set them equal to each other, \(\sqrt{x} = \sqrt{x + 1} + 8\), and solve for x. However, no solution exists for this equation. Hence, these two functions do not intersect.

Key concepts

Domain of a Function

When studying functions in algebra, one of the fundamental concepts is the domain of a function. The domain consists of all the possible input values (typically x-values) for which the function is defined. Understanding the domain is critical for sketching the graph of the function and for solving algebraic problems.

For example, the function given by the equation f(x) = \(\sqrt{x}\) only has real-number outputs when x is zero or a positive number. That's because the square root of a negative number isn't defined in the real number system. Therefore, the domain of f(x) is [0, +\infty), which means it includes all real numbers from 0 to positive infinity. This set includes 0 but does not include positive infinity since we can always find a larger number.

Similarly, the domain of the function g(x) = \(\sqrt{x+1}\) + 8 is determined by the condition that the expression under the square root, x+1, must be non-negative. This leads to the domain [-1, +\infty), signifying that g(x) is defined for all real numbers starting from -1 to positive infinity, inclusive of -1.

Plotting Functions

Visual representation of functions through plots is a great way to comprehend their behavior. Plotting functions on a coordinate plane involves drawing a curve or a line that represents all the points (x, f(x)) where f(x) is the value of the function at each x within its domain.

In the given example, plotting the function f(x) = \(\sqrt{x}\) would involve marking points on the graph starting from the origin (0,0) and moving rightward, since the domain starts from zero. The curve rises slowly because the square root increases at a decreasing rate as x gets larger.

On the other hand, plotting g(x) = \(\sqrt{x+1}\) + 8 requires a shift. The square root part will look similar to f(x), but it moves one unit left and eight units upward. These translations are due to the +1 inside the square root (shifts left) and the +8 outside of it (shifts up).

Intersection of Functions

An intersection of functions occurs at the points where two or more graphs coincide. In algebra, finding where the functions intersect can help us solve equations and understand relationships between different functions.

To find the intersection between f(x) = \(\sqrt{x}\) and g(x) = \(\sqrt{x+1}\) + 8, we set them equal to each other and solve for x. This process can sometimes reveal that no actual intersection points exist, as is the case here. The equation \(\sqrt{x} = \sqrt{x + 1} + 8\) implies that the square root of x would have to be 8 units larger than the square root of x+1. Since this isn't possible, it indicates that the graphs of f(x) and g(x) don't cross at any point.

This means there are no real solutions to the equation and, geometrically, the two functions do not intersect on the coordinate plane. Such an analysis is especially important in real-world scenarios where finding points of intersection often represent simultaneous solutions to related problems.