Problem 5 - Intermediate Algebra

Question

Refer to the graph on the right. GRAPH CANNOT COPY a. What type of function is\(f(x)=\log _{4} x ?\) b. What is the domain of the function? What is the range of the function? c. What is the \(y\) -intercept of the graph? What is the \(x\) -intercept of the graph? d. Is \(f\) a one-to-one function? e. What is an asymptote of the graph? I. Is \(f\) an increasing or a decreasing function? g. The graph passes through the point \((4, y) .\) What is \(y ?\)

Step-by-Step Solution

Answer

a. Logarithmic. b. Domain: \((0, \infty)\), Range: \((-\infty, \infty)\). c. \((1, 0)\). d. Yes. e. \(x = 0\). f. Increasing. g. \(y = 1\).

Step 1: Determine the Function Type

The function given is \(f(x) = \log_{4}x\). This is a logarithmic function, with base 4.

Step 2: Identify the Domain and Range

For \(f(x) = \log_{4}x\), the domain includes all positive real numbers \((0, \infty)\), while the range is all real numbers \((-\infty, \infty)\). This is because a logarithm is defined for positive real numbers and can take any real value as its result.

Step 3: Find the Intercepts

The \(y\)-intercept is the point where the graph crosses the \(y\)-axis. However, for logarithmic functions, \(f(x) = \log_{4}x\) doesn't have a \(y\)-intercept because it is undefined for \(x = 0\). The \(x\)-intercept is the point where \(y = 0\), so we set \(\log_{4}x = 0\), which implies \(x = 1\). Thus, the \(x\)-intercept is at \((1, 0)\).

Step 4: Determine if the Function is One-to-One

A function is one-to-one if each \(x\)-value maps to a unique \(y\)-value. The function \(f(x) = \log_{4}x\) is one-to-one because it passes the horizontal line test - any horizontal line will intersect the curve at most once.

Step 5: Identify the Asymptote

The vertical asymptote of \(f(x) = \log_{4}x\) is at \(x = 0\) since as \(x\) approaches 0, \(f(x)\) tend to negative infinity.

Step 6: Determine if the Function is Increasing or Decreasing

The function \(f(x) = \log_{4}x\) is increasing because as \(x\) increases, \(f(x)\) also increases.

Step 7: Find the Value of \(y\) at the Point (4, y)

To find \(y\) when \(x = 4\), we calculate \(y = \log_{4} 4\). Since \(4 = 4^1\), \(\log_{4}4 = 1\). Thus, \(y = 1\).

Key Concepts

Domain and Range

The domain and range of a logarithmic function are critical concepts to understand. For the function \(f(x) = \log_{4} x\), the domain consists of all positive real numbers, represented as \((0, \infty)\). This is because logarithms are only defined for positive inputs; they cannot accept zero or negative numbers as inputs. This constraint is common to all logarithmic functions, irrespective of their bases.
Understanding the range of \(f(x) = \log_{4} x\) is equally important. The range includes all real numbers, from \(-\infty\) to \(\infty\). This property reflects the fact that a logarithmic function can develop outputs spanning from negative infinity to positive infinity, as the input value increases from just above zero to infinity.

Intercepts in Graphing

Intercepts are the points where a graph crosses the x-axis or y-axis. For logarithmic functions, the topic of intercepts offers unique insights. With \(f(x) = \log_{4} x\), we face a situation where the function does not have a \(y\)-intercept. This occurs because the logarithmic function is undefined at \(x = 0\).
On the other hand, the \(x\)-intercept is at the point where \(y = 0\). To find this for \(f(x) = \log_{4} x\), set the equation to zero: \(\log_{4} x = 0\). Solving this equation shows that the function intersects the x-axis at \(x = 1\), thus the \(x\)-intercept is \((1, 0)\). This emphasizes an important property—logarithmic functions in their natural form do not cross the y-axis, instead they just meet the x-axis.

One-to-One Functions

When discussing logarithmic functions like \(f(x) = \log_{4} x\), it's essential to address whether the function is one-to-one. A one-to-one function ensures that each input \(x\) corresponds to a unique output \(y\).
This property is verified by the horizontal line test. A function is one-to-one if no horizontal line intersects the graph more than once. For \(f(x) = \log_{4} x\), each distinct \(x\)-value produces a unique \(y\)-value, making it a classic example of a one-to-one function. This unique pairing means that there are no repeated outputs for different inputs.

Asymptotes

Asymptotes are invisible lines that a graph approaches but never actually reaches. For \(f(x) = \log_{4} x\), an important point to note is the presence of a vertical asymptote at \(x = 0\).
When we say that there is a vertical asymptote at \(x = 0\), it means that as the input \(x\) gets exceedingly close to zero, the output \(f(x)\) plummets towards negative infinity. This behavior is characteristic of logarithmic functions, marking a border they can never touch or cross, reflecting their mathematical limitation of not accepting non-positive numbers.

Increasing and Decreasing Functions

Is the function increasing or decreasing? For \(f(x) = \log_{4} x\), the function is increasing. But what does that mean?
An increasing function is one where as \(x\) values rise, the \(y\) or function values also rise. For \(f(x) = \log_{4} x\), as \(x\) moves from values greater than 0 upwards, \(f(x)\) also consistently gets larger. This consistent increase is due to the positive base of the logarithm, which ensures that as you move right along the x-axis, the graph moves upwards.