Problem 31 - Calculus with Concepts in Calculus

Question

Find the horizontal asymptote of the graph of the function. Then sketch the graph of the function. $$ f(x)=\sqrt{\frac{3-x}{4-x}} $$

Step-by-Step Solution

Answer

The horizontal asymptote is \( y = 1 \).

Step 1: Understand the Concept of Horizontal Asymptotes

Horizontal asymptotes represent the behavior of a function as the input values approach infinity or negative infinity. For the function \( f(x) = \sqrt{\frac{3-x}{4-x}} \), we need to determine its behavior when \( x \rightarrow \infty \).

Step 2: Simplify the Expression as x Approaches Infinity

As \( x \rightarrow \infty \), the terms \( 3-x \) and \( 4-x \) both approximate \( -x \) since the constants become insignificant. This simplifies \( \frac{3-x}{4-x} \rightarrow \frac{-x}{-x} = 1 \). Thus, \( f(x) \approx \sqrt{1} = 1 \).

Step 3: Determine the Horizontal Asymptote

Since \( f(x) \rightarrow 1 \) as \( x \rightarrow \infty \), the horizontal asymptote of the function is \( y = 1 \).

Step 4: Understand the Restrictions for x-values

The function is only defined where the argument under the square root is non-negative. Set \( \frac{3-x}{4-x} \geq 0 \) to find valid values for \( x \). This inequality holds when \( x \leq 3 \) and \( x > 4 \), ensuring the function is not defined between these values.

Step 5: Sketch the Function

The graph approaches \( y = 1 \) as \( x \rightarrow \infty \) but is only defined for \( x \leq 3 \) and \( x > 4 \). Between these x-values, since the function is undefined, there is a gap in the graph. Reflect this behavior in the sketch of the function.

Key Concepts

Function Behavior

Understanding the behavior of a function is crucial to analyzing its graph and relating this to features like horizontal asymptotes. Function behavior refers to how the output of a function changes as the input, typically the variable \( x \), grows large or becomes very small.
In this particular function, \( f(x) = \sqrt{\frac{3-x}{4-x}} \), we are interested in what happens as \( x \to \infty \). The numerator and denominator \( 3-x \) and \( 4-x \) both trend towards negative infinity, making the behavior of these terms approachable as \( -x \).
Thus, the ratio \( \frac{3-x}{4-x} \) approximates \( \frac{-x}{-x} = 1 \) for very large \( x \). This simplification offers us insight into the function's horizontal asymptote at \( y = 1 \), indicating that the function value itself, \( f(x) \), tends towards 1 as \( x \) increases indefinitely.

Graph Sketching

When sketching the graph of a function like \( f(x) = \sqrt{\frac{3-x}{4-x}} \), start by identifying critical features like intercepts and asymptotes. More importantly, observe the continuous nature or any gaps where the function is not defined.
The horizontal asymptote at \( y = 1 \) plays a pivotal role. As \( x \) approaches infinity, \( f(x) \) nears this horizontal line, guiding how the graph unfolds far along the \( x \)-axis.
The function is not defined between \( x = 3 \) and \( x = 4 \), creating a gap on the graph. This results in two separate portions on the graph, one for \( x \leq 3 \) and another for \( x > 4 \). Highlight these intervals to ensure your sketch reflects these discontinuities. Connect these ideas when sketching by smoothly approaching the asymptote for very large or very small \( x \) and stopping abruptly at the endpoints where the function is undefined.

Inequalities in Functions

Inequalities help clarify where a function like \( f(x) = \sqrt{\frac{3-x}{4-x}} \) is actually defined. They determine valid \( x \)-values and ensure the function outputs real numbers.
To solve \( \frac{3-x}{4-x} \geq 0 \), apply critical point analysis, identifying breakpoints where the expression equals zero or becomes undefined. These arise when the numerator or denominator equals zero. Here, \( 3-x = 0 \) gives \( x = 3 \), and \( 4-x = 0 \) gives \( x = 4 \).

• When \( x = 3 \), the expression evaluates to \( 0 \), making it a potential endpoint.
• As \( x > 4 \), both terms are negative, yielding a positive interval.

This information shows that the function is defined for \( x \leq 3 \) and \( x > 4 \). Applying inequality solutions ensures you understand where the function exists, allowing for accurate graphical representation by excluding undefined regions between \( x = 3 \) and \( x = 4 \).