Let f(x) = 1/(7-sin 5x) be a function defined on R. Then the range of the function f(x) is equal to

Q: Wait, does the $5x$ inside the sine function change the range? Shouldn't it be different from $\sin x$?

That is a fantastic question! It is a common trap to think the $5x$ scales the range. However, the sine function, regardless of the argument (as long as it is a real number), always oscillates between $-1$ and $1$. The $5x$ only changes the *frequency* of the oscillation, not the *amplitude*. So, $\sin 5x$ still stays strictly within $[-1, 1]$.

Q: Why do the inequality signs flip when we take the reciprocal?

Think about it with numbers. We know $2 0.25$, the inequality must flip! Whenever you take the reciprocal of positive numbers, the order reverses because a larger denominator results in a smaller fraction. It is a crucial rule to remember.

Q: Why is the range $[1/8, 1/6]$ and not $[1/6, 1/8]$?

In mathematics, interval notation $[a, b]$ always requires $a \le b$. We must always list the lower bound first and the upper bound second. Since $1/8$ is smaller than $1/6$, the correct interval is $[1/8, 1/6]$.

Question

Let $f(x) = \frac{1}{7-\sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to ;

Options:

(A) $[\frac{1}{7}, \frac{1}{6}]$

(B) $[\frac{1}{8}, \frac{1}{5}]$

(C) $[\frac{1}{7}, \frac{1}{5}]$

(D) $[\frac{1}{8}, \frac{1}{6}]$

Sigma Percentile · Accepted Answer

### Analyzing the Function

We are tasked with finding the range of the function:

$$f(x) = \frac{1}{7 - \sin 5x}$$

When approaching a function like this, we treat it as a transformation process. Our goal is to determine all possible output values by analyzing the behavior of the core trigonometric component.

### Phase 1: The Oscillating Core

Every complex function has a heartbeat. In this case, the heartbeat is the trigonometric term $\sin 5x$.

We know that for any real number $	heta$, the sine function is strictly bounded:

$$-1 \le \sin 	heta \le 1$$

Even though our input is $5x$, the sine function remains constrained. Thus, we establish our foundation:

$$-1 \le \sin 5x \le 1$$

### Phase 2: Building the Denominator

Now, we transform this core into the denominator, $7 - \sin 5x$, by applying the laws of inequalities step-by-step. First, we negate the inequality:

$$-1 \le -\sin 5x \le 1$$

Next, we add $7$ to every part of the inequality to match the structure of our denominator:

$$7 - 1 \le 7 - \sin 5x \le 7 + 1$$

Simplifying this, we find that the denominator is trapped within a specific, positive range:

$$6 \le 7 - \sin 5x \le 8$$

This is a crucial realization, as it confirms the denominator is always positive and never zero.

### Phase 3: The Reciprocal Trap

To complete the function $f(x) = \frac{1}{7 - \sin 5x}$, we must take the reciprocal of the denominator. When taking the reciprocal of an inequality involving positive numbers, the direction of the inequality reverses because as the denominator increases, the value of the fraction decreases.

Applying this to our inequality $6 \le 7 - \sin 5x \le 8$, we obtain:

$$\frac{1}{6} \ge \frac{1}{7 - \sin 5x} \ge \frac{1}{8}$$

### Final Calculation

We have arrived at the inequality $\frac{1}{6} \ge f(x) \ge \frac{1}{8}$. To present this in standard interval notation, we rearrange the values from smallest to largest:

$$\frac{1}{8} \le f(x) \le \frac{1}{6}$$

This indicates that the function $f(x)$ oscillates perfectly between the horizontal lines $y = \frac{1}{8}$ and $y = \frac{1}{6}$.

The **range** of the function is the closed interval:

$$\left[ \frac{1}{8}, \frac{1}{6} ight]$$

Animated Solution for Mathematics - Functions: Let $f (x) = \frac{1}{7 - s i n 5 x}$ be a function defined on $R$ . Then the range of the function $f (x)$ is equal to ;

Select Answer:

Visualized Solution

Introduction to the Function

The Core Variable: $sin 5 x$

Range of Sine Function

Bounding $sin 5 x$

Transforming the Inequality

Building the Denominator

Bounds of the Denominator

The Reciprocal Transformation

Taking the Reciprocal

Rearranging the Bounds

Final Range of $f (x)$

The Sigma Insight: Domain and Range of a Function

Analyzing the Function

Phase 1

Phase 2

Phase 3

Final Calculation

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Animated Solution for Mathematics - Functions: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=7−sin5x1​ be a function defined on R. Then the range of the function f(x) is equal to ;