Given The Function $f:\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \rightarrow\left[\frac{1}{\sqrt{2}}, 1\right] ; F(x)=\cos X$, Check Which One(s) Of The Properties It Has.A. Injective B. Decreasing C. Surjective D. Strictly Decreasing E.

Properties of the Function $f(x) = \cos x$

The function $f(x) = \cos x$ is a fundamental trigonometric function that has been extensively studied in mathematics. In this article, we will investigate the properties of the function $f(x) = \cos x$ over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ and determine which of the given properties it satisfies.

The domain of the function $f(x) = \cos x$ is the set of all possible input values, which in this case is the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. The range of the function is the set of all possible output values, which is the interval $\left[\frac{1}{\sqrt{2}}, 1\right]$.

A function is said to be injective if it maps distinct elements of its domain to distinct elements of its range. In other words, if $f(x_1) = f(x_2)$, then $x_1 = x_2$. To determine if the function $f(x) = \cos x$ is injective, we need to check if it satisfies this condition.

Let $x_1$ and $x_2$ be two distinct elements of the domain $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. We need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$. Since $f(x) = \cos x$, we have:

$$\cos x_1 = \cos x_2$$

Using the identity $\cos x = \cos (-x)$, we can rewrite the above equation as:

$$\cos x_1 = \cos (-x_2)$$

Since the cosine function is an even function, we have:

$$\cos x_1 = \cos x_2$$

This implies that $x_1 = -x_2$. However, since $x_1$ and $x_2$ are both in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, we have $-\frac{\pi}{4} \leq x_1 \leq \frac{\pi}{4}$ and $-\frac{\pi}{4} \leq x_2 \leq \frac{\pi}{4}$. This implies that $x_1 = x_2$, which is a contradiction.

Therefore, the function $f(x) = \cos x$ is not injective over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

A function is said to be decreasing if its output value decreases as the input value increases. In other words, if $x_1 < x_2$, then $f(x_1) > f(x_2)$. To determine if the function $f(x) = \cos x$ is decreasing, we need to check if it satisfies this condition.

Let $x_1$ and $x_2$ be two elements of the domain $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ such that $x_1 < x_2$. We need to show that $f(x_1) > f(x_2)$. Since $f(x) = \cos x$, we have:

$$\cos x_1 > \cos x_2$$

Using the identity $\cos x = \cos (-x)$, we can rewrite the above equation as:

$$\cos x_1 > \cos (-x_2)$$

Since the cosine function is an even function, we have:

$$\cos x_1 > \cos x_2$$

This implies that $x_1 < x_2$ is not sufficient to guarantee that $f(x_1) > f(x_2)$. In fact, we can find counterexamples where $x_1 < x_2$ but $f(x_1) < f(x_2)$.

Therefore, the function $f(x) = \cos x$ is not decreasing over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

A function is said to be strictly decreasing if its output value decreases as the input value increases, and the output value is always less than the previous output value. In other words, if $x_1 < x_2$, then $f(x_1) > f(x_2)$ and $f(x_2) < f(x_1)$. To determine if the function $f(x) = \cos x$ is strictly decreasing, we need to check if it satisfies this condition.

Let $x_1$ and $x_2$ be two elements of the domain $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ such that $x_1 < x_2$. We need to show that $f(x_1) > f(x_2)$ and $f(x_2) < f(x_1)$. Since $f(x) = \cos x$, we have:

$$\cos x_1 > \cos x_2$$

Using the identity $\cos x = \cos (-x)$, we can rewrite the above equation as:

$$\cos x_1 > \cos (-x_2)$$

Since the cosine function is an even function, we have:

$$\cos x_1 > \cos x_2$$

However, we also have:

$$\cos x_2 > \cos x_1$$

This implies that $f(x_1) > f(x_2)$ and $f(x_2) < f(x_1)$, which is a contradiction.

Therefore, the function $f(x) = \cos x$ is not strictly decreasing over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

A function is said to be surjective if its range is equal to the codomain. In other words, for every element $y$ in the codomain, there exists an element $x$ in the domain such that $f(x) = y$. To determine if the function $f(x) = \cos x$ is surjective, we need to check if it satisfies this condition.

Let $y$ be an element of the codomain $\left[\frac{1}{\sqrt{2}}, 1\right]$. We need to show that there exists an element $x$ in the domain $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ such that $f(x) = y$. Since $f(x) = \cos x$, we have:

$$\cos x = y$$

Using the identity $\cos x = \cos (-x)$, we can rewrite the above equation as:

$$\cos x = \cos (-x)$$

Since the cosine function is an even function, we have:

$$\cos x = \cos x$$

This implies that $x = -x$, which is not possible since $x$ is in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. Therefore, we cannot find an element $x$ in the domain such that $f(x) = y$.

Therefore, the function $f(x) = \cos x$ is not surjective over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

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Q: What is the domain of the function $f(x) = \cos x$?

A: The domain of the function $f(x) = \cos x$ is the set of all possible input values, which in this case is the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$.

Q: What is the range of the function $f(x) = \cos x$?

A: The range of the function $f(x) = \cos x$ is the set of all possible output values, which is the interval $\left[\frac{1}{\sqrt{2}}, 1\right]$.

Q: Is the function $f(x) = \cos x$ injective?

A: No, the function $f(x) = \cos x$ is not injective over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. This means that there exist distinct elements $x_1$ and $x_2$ in the domain such that $f(x_1) = f(x_2)$.

Q: Is the function $f(x) = \cos x$ decreasing?

A: No, the function $f(x) = \cos x$ is not decreasing over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. This means that there exist elements $x_1$ and $x_2$ in the domain such that $x_1 < x_2$ but $f(x_1) > f(x_2)$.

Q: Is the function $f(x) = \cos x$ strictly decreasing?

A: No, the function $f(x) = \cos x$ is not strictly decreasing over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. This means that there exist elements $x_1$ and $x_2$ in the domain such that $x_1 < x_2$ but $f(x_1) > f(x_2)$ and $f(x_2) < f(x_1)$.

Q: Is the function $f(x) = \cos x$ surjective?

A: No, the function $f(x) = \cos x$ is not surjective over the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. This means that there exist elements $y$ in the codomain $\left[\frac{1}{\sqrt{2}}, 1\right]$ such that there is no element $x$ in the domain $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ such that $f(x) = y$.

Q: What are some common mistakes to avoid when working with the function $f(x) = \cos x$?

A: Some common mistakes to avoid when working with the function $f(x) = \cos x$ include:

• Assuming that the function is injective, decreasing, strictly decreasing, or surjective without checking the properties.
• Not considering the domain and range of the function.
• Not using the correct identity for the cosine function, such as $\cos x = \cos (-x)$.
• Not checking for counterexamples when trying to prove a property.

Q: How can I apply the properties of the function $f(x) = \cos x$ in real-world problems?

A: The properties of the function $f(x) = \cos x$ can be applied in real-world problems such as:

• Modeling the motion of a pendulum or a spring.
• Analyzing the behavior of a physical system that involves periodic motion.
• Designing a system that requires a specific range of values for a particular variable.
• Understanding the relationship between the input and output of a system.

Q: What are some other functions that have similar properties to the function $f(x) = \cos x$?

A: Some other functions that have similar properties to the function $f(x) = \cos x$ include:

• The sine function, $f(x) = \sin x$.
• The tangent function, $f(x) = \tan x$.
• The cotangent function, $f(x) = \cot x$.
• The secant function, $f(x) = \sec x$.
• The cosecant function, $f(x) = \csc x$.

These functions all have similar properties to the function $f(x) = \cos x$, such as being periodic, having a specific range, and being related to the cosine function.