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Magazine Chapter 2: Problem 7
Which of the following functions is one-to-one? (a) \(f(x)=x^{2}+4 x+3\) (b) \(g(x)=|x|+2\) (c) \(h(x)=\sqrt[3]{x+1}\) (d) \(F(x)=\cos x,-\pi \leq x \leq \pi\) (e) \(G(x)=e^{x}+e^{-x}\)
Short Answer
Expert verified
The function \(h(x) = \sqrt[3]{x+1}\) is one-to-one.
Step by step solution
01
Definition of a One-to-One Function
A function is defined as one-to-one (injective) if every element of the function's codomain is mapped to by at most one element of its domain. In other words, if \(f(a) = f(b)\) then \(a = b\). The function passes the horizontal line test; a horizontal line should intersect the graph of the function at most once.
02
Analyze Function (a)
The function \(f(x) = x^2 + 4x + 3\) is a quadratic function which is a parabola. Quadratic functions are not one-to-one because the horizontal line test will intersect the parabola at two points when not restricted to a domain.
03
Analyze Function (b)
The function \(g(x) = |x| + 2\) is an absolute value function shifted up by 2 units. Absolute value functions are not one-to-one because the horizontal line test will intersect the graph at two points: one in the positive direction and one in the negative direction of the x-axis.
04
Analyze Function (c)
The function \(h(x) = \sqrt[3]{x+1}\) is a cubic root function. Such functions are one-to-one because they pass the horizontal line test; they have an inverse over their entire domain.
05
Analyze Function (d)
The function \(F(x) = \cos x\) for \(-\pi \leq x \leq \pi\) is a trigonometric function. Cosine functions are not one-to-one over their standard intervals because they repeat values at different points within the interval.
06
Analyze Function (e)
The function \(G(x) = e^x + e^{-x}\) is known as a hyperbolic cosine function. The hyperbolic cosine function is not one-to-one because its minimum value occurs symmetrically around the y-axis, meaning the horizontal line test will intersect it at two locations for the same y-value.
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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 tool used to determine whether a function is one-to-one (injective). A function is considered one-to-one if each horizontal line intersects the graph at most once. This test helps us easily spot functions that don't have a unique output for each distinct input.
To perform the horizontal line test, imagine drawing horizontal lines across the graph of the function.
To perform the horizontal line test, imagine drawing horizontal lines across the graph of the function.
- If no horizontal line crosses the graph more than once, the function is injective, and it's safe to call it one-to-one.
- If any horizontal line crosses the graph more than once, then the function fails the test and is not injective.
Quadratic Functions
Quadratic functions like the one described by the formula \( f(x) = ax^2 + bx + c \) result in parabolas when graphed. The classic parabola shape is always opened upward or downward, depending on the sign of the coefficient \( a \).
An important property of quadratic functions is that they are not typically one-to-one. The horizontal line test will often show intersections at two points, meaning these functions do not satisfy the requirements to be injective.
An important property of quadratic functions is that they are not typically one-to-one. The horizontal line test will often show intersections at two points, meaning these functions do not satisfy the requirements to be injective.
- This is due to the symmetrical nature of parabolas about their vertex.
- Each value of \( f(x) \) could be achieved by two different values of \( x \) (one on each side of the vertex).
Cubic Root Functions
Cubic root functions, like \( h(x) = \sqrt[3]{x+1} \), produce graphs that curve smoothly through every part of their domain without creating symmetry that leads to repeated \( y \)-values. These functions are inherently one-to-one.
The graph of a cubic root function is characterized by a gentle "S" shape, passing through the origin and extending to infinity in both directions along the y-axis.
The graph of a cubic root function is characterized by a gentle "S" shape, passing through the origin and extending to infinity in both directions along the y-axis.
- They pass the horizontal line test easily because they have no repeated outputs for different inputs.
- This unique trait means they possess both a leftward and rightward movement without reversing direction and never converging onto a single line across its range.
Absolute Value Functions
Absolute value functions, such as \( g(x) = |x| + k \), graph into a "V" shape. This shape results from the absolute value operation, which makes all outputs non-negative, flipping any negative inputs to become positive.
These functions are not injective because they do not pass the horizontal line test.
These functions are not injective because they do not pass the horizontal line test.
- A horizontal line typically intersects an absolute value graph at two points, one on the positive side and one on the negative side of the x-axis.
- This means different x-values, one positive and one negative, map to the same y-value, violating the requirement for a function to be one-to-one.
