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Chapter 9: Problem 170

Show that \(f(a)=f(-a)\) if \(f(x)=x^{2}+3\)

Short Answer

Expert verified
To show that \(f(a) = f(-a)\) for the given function \(f(x) = x^2 + 3\), we substitute a and -a into the function and compare the resulting expressions. We find that \(f(a) = a^2 + 3\) and \(f(-a) = (-a)^2 + 3 = a^2 + 3\) (since \((-a)^2 = a^2\)). Since both expressions are equal, we have shown that \(f(a) = f(-a)\) for the given function.

Step by step solution

01

Write down the given function

We are given the function f(x) = x^2 + 3.
02

Substitute a into the function

We need to find f(a), so let's substitute a into the function: \(f(a) = a^2 + 3\).
03

Substitute -a into the function

Now, we need to find f(-a), so let's substitute -a into the function: \(f(-a) = (-a)^2 + 3\).
04

Simplify f(-a)

Let's simplify the expression we got in Step 3: \(f(-a) = (-a)^2 + 3 = a^2 + 3\) (since \((-a)^2 = a^2\)).
05

Compare f(a) and f(-a)

We found in Step 2 that \(f(a) = a^2 + 3\) and in Step 4 that \(f(-a) = a^2 + 3\). Since both expressions are equal, we have shown that \(f(a) = f(-a)\) for the given function \(f(x) = x^2 + 3\).

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Key Concepts

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

Function Symmetry
When we talk about function symmetry, we are discussing whether or not a function looks the same when it is flipped, either over the y-axis or other axes. Symmetry is like a mirror image and can help us understand a function's behavior more intuitively.

There are several types of symmetry that functions can have:
  • **Y-axis symmetry**: This occurs when the function graph looks the same on both sides of the y-axis. For a function, this means if you plug in positive and negative values of the same number, like x and -x, the function will output the same result. This is a hallmark of even functions.
  • **Origin symmetry**: For this type, if you rotate the graph 180 degrees around the origin, it would look the same. Functions with this kind of symmetry are odd functions.
  • **No symmetry**: Many functions do not exhibit symmetry over any axis or the origin.
For the function in our example, the symmetry is demonstrated by showing that substituting both a and -a results in the same output, confirming that the function is symmetrically mirrored over the y-axis.
Quadratic Functions
Quadratic functions are a type of polynomial function characterized by the highest power of the variable being 2. They can be written in the form:
\( f(x) = ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.

Quadratic functions have a distinct U-shaped graph called a parabola. Some key characteristics of quadratic functions include:
  • **Vertex**: The point where the parabola turns. It can be the lowest or highest point depending on whether the parabola opens upwards or downwards.
  • **Axis of symmetry**: A vertical line through the vertex that divides the parabola into two mirror-image halves.
  • **Roots/Zeroes**: Points where the parabola intersects the x-axis. Not all quadratic functions have real roots.
In the context of our example, \( f(x) = x^2 + 3 \), there's no linear term \( b \), which simplifies the function. The graph is symmetric around the y-axis, and this property supports what we proved: if you plug in positive and negative versions of the same number, the output remains the same.
Parity of Functions
The parity of functions refers to whether a function is even, odd, or neither. Understanding the parity is helpful in determining how a function acts over different parts of its domain.

Here's what you need to know about even and odd functions:
  • **Even functions**: A function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \) in the domain. Graphically, this means the function has y-axis symmetry. The example from the task, \( f(x) = x^2 + 3 \), is an even function because substituting \( x \) with \( -x \) gives the same value.
  • **Odd functions**: A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain. This property means that the graph of the function is symmetric around the origin, resembling a rotational symmetry.
  • **Neither**: Some functions do not fit either category.
Checking parity is usually straightforward but provides a powerful insight into the geometric nature of the function. In learning contexts, this allows students to anticipate the function's behavior, even without graphing.

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Most popular questions from this chapter

If \(\mathrm{f}(\mathrm{x})=(\mathrm{x}-2) /(\mathrm{x}+1)\), find the function values \(\mathrm{f}(2)\) \(\mathrm{f}(1 / 2)\), and \(\mathrm{f}(-3 / 4)\)

Find the relation defined by \(\mathrm{y}^{2}=25-\mathrm{x}^{2}\), where \(\mathrm{x}\) belongs to \(\mathrm{d}=\\{0,3,4,5\\}\).

Find the relation \(Q\) over \(\mathrm{S} \times \mathrm{T}\) if \(\mathrm{S}=\\{1,2,3\\}, \mathrm{T}=\\{4,5\\}\), and the rule of correspondence is \(\mathrm{r}(\mathrm{x})=\mathrm{x}+2\).

Let \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{2}\) with domain \(\mathrm{D}_{\mathrm{f}}=\mathrm{R}\) (or, alternatively, C) and \(\mathrm{g}(\mathrm{x})=\mathrm{x}-5\) with \(\mathrm{D}_{\mathrm{g}}=\mathrm{R}\) (or \(\left.\mathrm{C}\right)\) Find (a) \(\mathrm{f}+\mathrm{g} \quad\) (b) \(\mathrm{f}-\mathrm{g}\) (c) fg (d) \(\mathrm{f} / \mathrm{g}\).

Find the set of ordered pairs \(\\{(x, y)\\}\) if \(y=x^{2}-2 x-3\) and \(\mathrm{D}=\\{\mathrm{x} \mid \mathrm{x}\) is an integer and \(1 \leq \mathrm{x} \leq 4\\}\).
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