Problem 9 - Precalculus with Calculus Previews

Question

Sketch the graph of the given function \(f\). Find the \(y\) -intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing. $$ f(x)=3-\left(\frac{1}{5}\right)^{x} $$

Step-by-Step Solution

Answer

The function is increasing with \( y \)-intercept at \((0, 2)\) and horizontal asymptote \( y = 3 \).

Step 1: Understanding the Function

The given function is \( f(x) = 3 - \left(\frac{1}{5}\right)^x \). This is an exponential function where the base \( \frac{1}{5} \) is less than 1, indicating that it's a decreasing function.

Step 2: Finding the y-intercept

To find the \( y \)-intercept, substitute \( x = 0 \) into the function: \( f(0) = 3 - \left(\frac{1}{5}\right)^0 = 3 - 1 = 2 \). Therefore, the \( y \)-intercept is at point \((0, 2)\).

Step 3: Determining the Horizontal Asymptote

For the function \( f(x) = 3 - \left(\frac{1}{5}\right)^x \), as \( x \) approaches infinity, \( \left(\frac{1}{5}\right)^x \) approaches 0, making \( f(x) \) approach \( 3 \). Therefore, the horizontal asymptote is \( y = 3 \).

Step 4: Checking if the Function is Increasing or Decreasing

Since \( \frac{1}{5} \) is less than 1, the term \( \left(\frac{1}{5}\right)^x \) decreases as \( x \) increases. Therefore, \( f(x) = 3 - \left(\frac{1}{5}\right)^x \) increases as \( x \) increases.

Step 5: Sketching the Graph

The graph has a \( y \)-intercept at \( (0, 2) \) and a horizontal asymptote at \( y = 3 \). The function is increasing as \( x \) increases. The sketch should show the graph starting below \( y = 3 \) and increasing towards \( y = 3 \), without ever touching it.

Key Concepts

Horizontal Asymptotes

In the realm of exponential functions, horizontal asymptotes are lines that the graph approaches but never actually meets as the input values become very large or very small. For the function given in our problem, which is \( f(x) = 3 - \left(\frac{1}{5}\right)^x \), examining what happens as \( x \to \infty \) helps us understand where the horizontal asymptote lies. As \( x \) increases indefinitely, the term \( \left(\frac{1}{5}\right)^x \) diminishes towards zero because \( \frac{1}{5}^x \) represents increasingly small fractions. This makes our function's value \( f(x) \) tend ever closer to 3. Hence, the horizontal asymptote of this exponential function is \( y = 3 \). Remember, the horizontal asymptote tells us about the long-term behavior of the function, no matter how far into the positive or negative direction you look, the graph will hover around this line.

Y-intercepts

The \( y \)-intercept of a function is where the graph crosses the \( y \)-axis. We find it by substituting \( x = 0 \) into the function's equation. For our exponential function, \( f(x) = 3 - \left(\frac{1}{5}\right)^x \), substituting \( x = 0 \) gives:\[ f(0) = 3 - \left(\frac{1}{5}\right)^0 = 3 - 1 = 2 \]Thus, the \( y \)-intercept is at the point \((0, 2)\). This means that when \( x = 0 \), the value of the function is 2, and the graph of the function will pass through the \( y \)-axis at this point.Knowing the \( y \)-intercept is helpful when sketching the initial behavior of the graph, as it gives a definitive point the function must pass through.

Increasing and Decreasing Functions

Whether a function is increasing or decreasing is crucial in understanding its behavior. For exponential functions like \( f(x) = 3 - \left(\frac{1}{5}\right)^x \), this is determined by the base of the exponential term. If the base of the exponential is between 0 and 1, like \( \frac{1}{5} \), the function component \( \left(\frac{1}{5}\right)^x \) decreases as \( x \) increases. This makes our entire function \( f(x) = 3 - \left(\frac{1}{5}\right)^x \) increase as the \( \left(\frac{1}{5}\right)^x \) term decreases and gets subtracted from a constant value (3), leading to increasing values of \( f(x) \). In simple terms:- As \( x \) gets larger, \( \left(\frac{1}{5}\right)^x \) gets smaller.- Subtracting a smaller number from 3 makes \( f(x) \) larger.Therefore, \( f(x) \) is an increasing function in this context, and understanding this helps in predicting how the graph behaves throughout its range.