If G(x) = 4x - 7, what does G(1) represent?

G(1) is the output of the function when the input x is 1, which is -3.

What is the process to compute G(1) given G(x) = 4x - 7?

Insert x = 1 into the function: G(1) = 4(1) - 7, then simplify to get -3.

How does the value of G(1) change if the function is G(x) = 4x - 7?

For x = 1, G(1) is fixed at -3; changing the function would alter this value.

If G(x) = 4x-7 Then G(1) = ?

Q: How do you evaluate G(1) when G(x) = 4x - 7?

Substitute x = 1 into the function: G(1) = 4(1) - 7, which simplifies to -3.

If G(x) = 4x - 7 Then G(1) = ?

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Understanding the Function G(x) = 4x - 7

Introduction to Functions

A function is a mathematical relationship that assigns each input to exactly one output. In the notation G(x), G is the function name, and x is the variable or input. The expression following the equal sign defines how to compute the output based on the input x.

Analyzing the Given Function

The function in question is G(x) = 4x - 7. This is a linear function, meaning its graph is a straight line. The function involves multiplying the input x by 4 and then subtracting 7 from the result.

Key Components of the Function

Coefficient of x (4): Indicates the slope of the line, showing how much G(x) changes for each unit increase in x.
Constant term (-7): Represents the y-intercept, the point where the line crosses the y-axis when x = 0.

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Calculating G(1): Step-by-Step Process

Substituting the Value of x

To find G(1), substitute x = 1 into the function: \[ G(1) = 4(1) - 7 \] which simplifies to: \[ G(1) = 4 - 7 \]

Performing the Arithmetic

The calculation involves basic arithmetic:

Multiply 4 by 1, which gives 4.
Subtract 7 from 4, resulting in -3.

Result of the Calculation

Thus, the value of G(1) is: \[ G(1) = -3 \]

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Understanding the Significance of G(1) = -3

Interpretation in the Context of the Function

The value G(1) = -3 indicates that when the input x is 1, the output of the function is -3. In graph terms, the point (1, -3) lies on the line defined by G(x).

Graphical Representation

The line crosses the y-axis at -7 (the y-intercept).
When x increases by 1, G(x) increases by 4 (the slope).

Implications of the Result

This calculation exemplifies how linear functions behave and how specific inputs translate to outputs.

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Broader Context and Applications of Such Functions

Linear Functions in Real Life

Linear functions like G(x) = 4x - 7 appear in numerous real-world scenarios, including:

Calculating total cost based on the number of items purchased.
Estimating distance traveled over time with constant speed.
Modeling relationships where change is proportional and consistent.

Understanding the Slope and Intercept

The slope (4) suggests a steady increase in the output as the input increases.
The y-intercept (-7) indicates the starting point when x is zero.

Examples of Practical Usage

Cost calculation: If each item costs $4 and there's a fixed fee of $7, then total cost for x items is G(x) = 4x - 7.

Physics: Distance traveled with constant velocity over time can be modeled similarly.

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Extending the Concept: Other Inputs and Outputs

Calculating G(0)

Substitute x = 0: \[ G(0) = 4(0) - 7 = -7 \] This confirms the y-intercept point at (0, -7).

Calculating G(2)

Substitute x = 2: \[ G(2) = 4(2) - 7 = 8 - 7 = 1 \] The point (2, 1) lies on the line.

General Pattern

For any input x, the output can be found using: \[ G(x) = 4x - 7 \] which demonstrates the linear relationship between input and output.

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Understanding the Impact of Changing the Function

Adjusting the Coefficients

Changing the coefficient of x affects the slope and the steepness of the line.
Modifying the constant term shifts the line vertically.

Example: G(x) = 3x + 2

Slope: 3 (less steep)
Y-intercept: 2
For x = 1:

\[ G(1) = 3(1) + 2 = 5 \]

Comparison with Original Function

G(x) = 4x - 7 yields G(1) = -3.
G(x) = 3x + 2 yields G(1) = 5.

This illustrates how changing parameters influences the output at specific points.

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Summary and Conclusion

Recap of the Calculation

Given the function G(x) = 4x - 7.
To find G(1), substitute x = 1.
Calculation: G(1) = 4(1) - 7 = -3.
Therefore, G(1) = -3.

Significance of the Result

Understanding how to evaluate the function at specific points helps in graphing, interpreting, and applying linear functions across various disciplines.

Final Remarks

Mastering the process of substituting values into functions and performing basic arithmetic is fundamental in algebra. Recognizing the structure of linear functions enables students and practitioners to model real-world situations effectively, analyze relationships, and make informed predictions.

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In essence, the value of G(1) in the function G(x) = 4x - 7 is -3, illustrating the straightforward yet powerful nature of linear functions and their applications.