Question

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=x+\frac{1}{\sqrt{x}} ;\) shifted horizontally to the left by 3 units

Short answer

The function for the transformed graph after shifting horizontally to the left by 3 units is \(g(x) = x+3 + \frac{1}{\sqrt{x+3}}\).

Step-by-step solution

Identify the given function

The given function is \(f(x) = x+\frac{1}{\sqrt{x}}\).

Shift the graph horizontally to the left by 3 units

To shift the graph horizontally to the left by 3 units, replace \(x\) with \(x+3\) in the function:\(f(x+3) = (x+3) + \frac{1}{\sqrt{x+3}}\).

Simplifying the transformed function

The transformed function can be simplified to:\(g(x) = x+3 + \frac{1}{\sqrt{x+3}}\). Therefore, the function for the transformed graph after shifting horizontally to the left by 3 units is \(g(x) = x+3 + \frac{1}{\sqrt{x+3}}\).

Key concepts

Horizontal Shifts

Understanding horizontal shifts is crucial when it comes to manipulating the graph of a function. When a function undergoes a horizontal shift, this means that every point on the graph moves to the left or right along the x-axis.

A shift to the right is represented by a subtraction within the function's argument, while a shift to the left is indicated by an addition. For instance, a function described as \( f(x) \) that shifts 3 units to the left will be transformed into \( f(x+3) \).

Real-World Example of Horizontal Shifts

Imagine you are looking at a chart that shows the growth of a plant over time. If you wanted to represent that the plant started growing three days earlier, you would shift the entire growth chart three units to the left on the time axis. Similarly, in function transformations, shifting a graph horizontally to the left by 3 units would mean replacing every occurrence of \( x \) with \( x+3 \) in the function's formula.

Function Transformation

Function transformation entails making changes to the original function's formula to produce a variation in its graph. Transformations can include shifts, reflections, stretching, and compressing the graph. Essentially, it modifies the position or shape of the graph without changing the relationship between the inputs (x-values) and the outputs (y-values) of the function.

In mathematical terms, transforming a function \( f(x) \) can involve various operations such as \( f(x+a) \), \( f(x) + b \), \( cf(x) \), and \( f(-x) \), where \( a \) and \( b \) are constants that translate the graph horizontally and vertically respectively, and \( c \) is a constant that stretches or compresses the graph.

Key Points in Function Transformation

• Horizontal and vertical shifts do not alter the shape of the graph, only its position.
• Stretching and compressing change the graph's steepness or width but maintain its general pattern.
• Reflections flip the graph over a specified axis, creating a mirror image.

Understanding these transformations allows for quick and efficient manipulation of graphs to better analyze or visualize the underlying relationships in data or equations.

Graph of a Function

The graph of a function is a visual representation that shows all the possible (x, y) pairs that satisfy the function. It is a powerful tool for visual learners, as it allows them to see the behavior and features of functions such as intercepts, maxima, minima, and asymptotes.

To graph a function, you plot on the coordinate plane the points corresponding to the inputs from the domain and their respective outputs from the range. The graph can be transformed following different algebraic manipulations to the function's equation.

Interpreting Graphs

For instance, when you graph \( f(x) = x + \frac{1}{\sqrt{x}} \) and shift it horizontally to the left, you are, in effect, seeing how the function behaves when you alter its 'starting point' on the x-axis. Graphing functions can uncover the properties of the equation that might not be immediately evident from the algebraic form alone, such as periodicity, symmetry, and even continuity.

Whether you are learning the basics of algebra or delving into complex analysis, the ability to comprehend and manipulate the graph of a function is an essential skill in mathematics.