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Chapter 2: Problem 18

Determine whether the function is one-to-one. $$h(x)=x^{3}+8$$

Short Answer

Expert verified
The function \( h(x) = x^3 + 8 \) is one-to-one.

Step by step solution

01

Understand the definition of a one-to-one function

A function is one-to-one (injective) if, for every pair of different inputs, the outputs are different. In other words, if \( f(a) = f(b) \), then \( a = b \). Another way to determine if a function is one-to-one is to check if it passes the horizontal line test on a graph.
02

Analyze the function algebraically

To determine algebraically if the function \( h(x) = x^3 + 8 \) is one-to-one, suppose \( h(a) = h(b) \). This implies that \( a^3 + 8 = b^3 + 8 \). By subtracting 8 from both sides, we have \( a^3 = b^3 \).
03

Solve the equation for variable

Given \( a^3 = b^3 \), take the cube root of both sides to get \( a = b \). Since this holds true for all values of \( a \) and \( b \), it suggests that the function is injective, validating that different inputs result in different outputs.
04

Conclusion

Since the condition \( a = b \) holds whenever \( h(a) = h(b) \), the function \( h(x) = x^3 + 8 \) is one-to-one. Additionally, if we plot the graph of \( h(x) \), it will pass the horizontal line test as each horizontal line intersects the graph at most once.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Line Test
The horizontal line test is a simple yet powerful visual method to determine if a function is one-to-one (also known as injective). When we plot the graph of a function, a horizontal line test involves drawing a horizontal line (parallel to the x-axis) across the graph.
  • If any horizontal line touches the graph in more than one place, the function fails the test and is not one-to-one.
  • If every horizontal line touches the graph at most once, the function is one-to-one.
For the function given, \( h(x) = x^3 + 8 \), its graph demonstrates that no horizontal line will ever intersect more than once, confirming it is indeed a one-to-one function according to this test.
Injective Function
An injective function, commonly termed as one-to-one, ensures that each input maps to a unique output. This implies that if two different inputs are plugged into the function, they'll yield distinct results.
To determine if a function is injective:
  • Assume \( f(a) = f(b) \).
  • Prove that this assumption leads to \( a = b \).
For the expression \( h(x) = x^3 + 8 \), if \( h(a) = h(b) \), then \( a^3 + 8 = b^3 + 8 \). By stripping away the constant (subtracting 8), we are left with \( a^3 = b^3 \). Solving this shows \( a = b \), confirming its injectivity as different x-values will always result in different y-values.
Algebraic Analysis
Algebraic analysis provides a concrete approach to determine whether a function is injective without graphing. With this method, you typically solve the function's equation algebraically to ascertain that each input has a unique output.
For \( h(x) = x^3 + 8 \), testing for injectivity means solving this equation for \( a \) and \( b \):
  • \( h(a) = h(b) \) becomes \( a^3 + 8 = b^3 + 8 \).
  • Subtracting 8 gives \( a^3 = b^3 \).
  • Taking the cube root on both sides yields \( a = b \).
This demonstrates that the equality of images implies the equality of inputs, verifying that \( h(x) \) is indeed an injective function. In essence, any pair of distinct inputs will produce distinct outputs. This is a key characteristic of one-to-one functions and is essential for confirming their injectivity through algebraic methods.

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Most popular questions from this chapter

DISCOVER: Graph of the Absolute Value of a Function (a) Draw graphs of the functions $$f(x)=x^{2}+x-6$$ $$\text { and } \quad g(x)=\left|x^{2}+x-6\right|$$ How are the graphs of \(f\) and \(g\) related? (b) Draw graphs of the functions \(f(x)=x^{4}-6 x^{2}\) and \(g(x)=\left|x^{4}-6 x^{2}\right| .\) How are the graphs of \(f\) and \(g\) related? (c) In general, if \(g(x)=|f(x)|,\) how are the graphs of \(f\) and \(g\) related? Draw graphs to illustrate your answer.

Graphing Functions Sketch a graph of the function by first making a table of values. $$g(x)=x^{2}+2 x+1$$

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} -1 & \text { if } x < -1 \\ x & \text { if }-1 \leq x \leq 1 \\ 1 & \text { if } x > 1 \end{array}\right.$$

DISCUSS: Even and Odd Power Functions What must be true about the integer \(n\) if the function $$ f(x)=x^{n} $$ is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?

DISCUSS: Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.
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