Let $f$ Be A Linear Function Such That: $f(1) = 3$. 1. A - Determine The Coefficient Of The Function $f$. B - Deduce The Expression Of $f$. 2. Calculate The Image Of $-2$; $3$ And $\frac{5}{3}$.

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In mathematics, linear functions form the bedrock of numerous concepts and applications. They are characterized by their straightforward structure and predictable behavior, making them invaluable tools in modeling real-world phenomena. This article delves into the intricacies of a specific linear function, f, defined by the condition f(1) = 3. We will meticulously determine the coefficient of this function, derive its explicit expression, and subsequently calculate the images of various inputs, namely -2, 3, and 5/3. By dissecting this problem, we gain a deeper understanding of the fundamental properties of linear functions and their practical significance.

1. Determining the Coefficient and Expression of the Linear Function f

1. a - Determining the Coefficient of the Function f

To kickstart our exploration, we first need to determine the coefficient of the linear function f. Recall that a linear function can be generally expressed in the form f(x) = ax, where a represents the coefficient, also known as the slope of the line. This coefficient dictates the rate at which the function's output changes with respect to its input. The given condition, f(1) = 3, provides a crucial piece of information that allows us to pinpoint the value of a. By substituting x = 1 into the general form of the linear function, we obtain f(1) = a(1) = a. Since we know that f(1) = 3, it directly follows that a = 3. Therefore, the coefficient of the linear function f is 3. Understanding this coefficient is paramount as it dictates the function's steepness and direction. A larger coefficient implies a steeper slope, while a negative coefficient indicates a decreasing function. In our case, the positive coefficient of 3 signifies that the function f is increasing, meaning that its output values increase as the input values increase. This initial step of determining the coefficient lays the foundation for understanding the behavior and characteristics of the linear function f.

1. b - Deducing the Expression of f

Now that we've successfully determined the coefficient of the linear function f to be 3, we can proceed to deduce the explicit expression for f(x). As we established earlier, a linear function takes the general form f(x) = ax. Having found that a = 3, we can directly substitute this value into the general form. This substitution yields f(x) = 3x. This simple yet powerful expression encapsulates the essence of the linear function f. It dictates that for any given input x, the function's output is simply three times that input. This expression serves as a versatile tool for evaluating the function's output for any value of x. It also provides a clear and concise representation of the function's behavior. For every unit increase in x, the function's output f(x) increases by three units. This constant rate of change is a hallmark of linear functions and is directly reflected in the coefficient 3. The expression f(x) = 3x not only allows us to compute specific output values but also provides a global understanding of the function's behavior across its entire domain. This understanding is crucial for various applications, including modeling linear relationships in real-world scenarios.

2. Calculating the Image of -2, 3, and 5/3

Having successfully determined the expression for the linear function f as f(x) = 3x, we can now proceed to calculate the images of specific input values. The image of a value under a function is simply the output of the function when that value is used as the input. We will focus on calculating the images of -2, 3, and 5/3. These values are strategically chosen to showcase the function's behavior for both negative and positive inputs, as well as for integer and fractional inputs. This comprehensive approach will provide a well-rounded understanding of how the function operates across different input ranges. By calculating these images, we gain concrete examples of the function's output for various inputs, further solidifying our grasp of its linear nature. This process is not only mathematically valuable but also crucial for applying the function in practical scenarios, where specific input-output relationships need to be understood.

2. a - Calculating the Image of -2

To calculate the image of -2 under the linear function f, we simply substitute x = -2 into the expression f(x) = 3x. This yields f(-2) = 3*(-2) = -6. Therefore, the image of -2 under the function f is -6. This result signifies that when the input is -2, the function's output is -6. This negative output is consistent with the negative input and the positive coefficient of the function. The function essentially scales the input by a factor of 3, and since the input is negative, the output is also negative. This calculation exemplifies the linear relationship between input and output in the function f. The magnitude of the output (-6) is three times the magnitude of the input (-2), reflecting the coefficient of 3. This straightforward calculation highlights the predictable nature of linear functions. For any given input, the output can be easily determined by simply multiplying the input by the coefficient. Understanding this relationship is fundamental for analyzing and applying linear functions in various mathematical and real-world contexts.

2. b - Calculating the Image of 3

Next, let's calculate the image of 3 under the linear function f. As before, we substitute x = 3 into the expression f(x) = 3x. This gives us f(3) = 3*(3) = 9. Consequently, the image of 3 under the function f is 9. This result demonstrates the function's behavior for a positive input. When the input is 3, the function's output is 9. This positive output aligns with the positive input and the positive coefficient of the function. The function scales the input by a factor of 3, resulting in an output that is three times the input. This calculation further reinforces the linear relationship between input and output in the function f. The output (9) is directly proportional to the input (3), with the constant of proportionality being the coefficient 3. This direct proportionality is a defining characteristic of linear functions and allows for easy prediction of output values for any given input. This calculation provides another concrete example of how the function f operates and solidifies our understanding of its linear nature.

2. c - Calculating the Image of 5/3

Finally, we will calculate the image of 5/3 under the linear function f. Substituting x = 5/3 into the expression f(x) = 3x, we obtain f(5/3) = 3*(5/3) = 5. Therefore, the image of 5/3 under the function f is 5. This calculation demonstrates the function's behavior for a fractional input. When the input is 5/3, the function's output is 5. This result showcases how the function scales fractional inputs in a consistent manner. The function multiplies the input by 3, effectively canceling out the denominator in the fraction and resulting in a whole number output. This calculation highlights the versatility of linear functions in handling different types of inputs. Whether the input is an integer, a fraction, or any other real number, the function f applies the same linear transformation, multiplying it by the coefficient 3. This consistency is a key feature of linear functions and makes them widely applicable in various mathematical and scientific domains. The image of 5/3 being 5 further solidifies our understanding of the function's behavior and its ability to handle fractional inputs seamlessly.

In conclusion, this exploration of the linear function f defined by f(1) = 3 has provided a comprehensive understanding of its properties and behavior. We successfully determined the coefficient of the function to be 3, deduced its expression as f(x) = 3x, and calculated the images of -2, 3, and 5/3 to be -6, 9, and 5, respectively. These calculations have reinforced the fundamental characteristics of linear functions, including their constant rate of change and direct proportionality between input and output. The ability to determine the coefficient, derive the expression, and calculate images is crucial for applying linear functions in various mathematical and real-world contexts. This analysis serves as a valuable foundation for further exploration of more complex mathematical concepts and their applications.