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Magazine Chapter 11: Problem 57
Graph the given functions. $$f(t)=t^{-4 / 5}$$
Short Answer
Expert verified
The function \( f(t)=t^{-4/5} \) is symmetric about the y-axis, undefined at \( t=0 \), and approaches zero as \( t \to \pm\infty \).
Step by step solution
01
Analyze the Function Type
The function given is a power function, represented as \( f(t) = t^{-4/5} \). This type of function includes a variable raised to a fractional exponent. The negative exponent indicates a reciprocal relationship, meaning \( f(t) = \frac{1}{t^{4/5}} \).
02
Identify the Domain
Since the function involves a negative exponent, it's equivalent to a reciprocal, which is defined for all non-zero values of \( t \). Therefore, the domain is all real numbers \( t eq 0 \).
03
Determine Symmetry and Behavior
The function \( f(t) = t^{-4/5} \) is an even function because \( f(t) = f(-t) \). This property implies it's symmetric about the y-axis. As \( t \to 0^+ \) or \( t \to 0^- \), \( f(t) \to \,\infty \) but diverges to large negative values as \( t \to \,\pm \,\infty \).
04
Sketching the Graph
Start with the behavior near \( t = 0 \), where the function rises steeply. Because of the negative exponent, as \( t \) grows larger in the positive or negative direction, \( f(t) \) decreases towards zero. Plot these general behaviors, ensuring the vertical asymptote at \( t = 0 \) and the symmetry. The graph should hug the axes at large positive and negative \( t \).
05
Use Key Points to Guide the Sketch
To assist with accuracy, pick specific points like \( t = 1 \) and \( t = -1 \). For \( t = 1 \), \( f(1) = 1^{-4/5} = 1 \). For \( t = -1 \), \( f(-1) = (-1)^{-4/5} = 1 \), supporting the even symmetry. These show precise points to graph, with the function nesting the horizontal axis further from the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fractional Exponents
Fractional exponents are a way to express powers and roots in a compact form. When you see an expression like \( t^{-4/5} \), it has two components of meaning: the numerator of the fraction indicates the power, and the denominator indicates the root.
- The exponent \(-4/5\) specifically means that the variable \( t \) is to be raised to the power of 4 (because of the 4 in the numerator) and then taken as the 5th root (because of the 5 in the denominator).
- The negative sign (")") in the exponent indicates that you take the reciprocal of the base raised to the positve power: \( \frac{1}{t^{4/5}} \).
Domain of a Function
The domain of a function refers to all the possible input values (usually denoted as \( t \) or \( x \)) that will produce a valid output. For the function \( f(t) = t^{-4/5} \), the key concern is the term \( \frac{1}{t^{4/5}} \).
- We need to avoid division by zero, which would make the expression undefined. Hence, \( t \) must not be zero.
- Since there are no other restrictions such as square roots of negatives in the real number system or logarithms, \( t \) can take any real value as long as it's not zero.
Even Functions
An even function is one where \( f(t) = f(-t) \) for all \( t \) in the function's domain. This means the function is symmetric about the y-axis. For \( f(t) = t^{-4/5} \), this holds true because replacing \( t \) with \( -t \) does not change the value of the function: \[f(-t) = (-t)^{-4/5} = (t)^{-4/5} = f(t).\]
- The symmetry property simplifies analyzing the graph: once you sketch the function for positive \( t \), you can mirror it exactly for negative \( t \).
- This property also assists in identifying key characteristics like intercepts and asymptotes since they will also be symmetric about the y-axis.
Graphing Functions
When graphing a function like \( f(t) = t^{-4/5} \), understanding its behavior is crucial. Let's break down how to plot this function:
- First, acknowledge the vertical asymptote at \( t = 0 \). The function rises towards infinity as \( t \) approaches zero from either side, due to the reciprocal of \( t^{4/5} \).
- For large positive and negative values of \( t \), the function value approaches zero. This is because as \( t^{4/5} \) becomes very large in the denominator, \( \frac{1}{t^{4/5}} \) becomes very small.
- Select easy-to-calculate points for your sketch, such as \( t = 1 \) and \( t = -1 \). At both of these, \( f(t) = 1 \), reaffirming symmetry.
