Let's analyze the functions step by step.
First set of functions:
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Calculating \( f(3) \): \[ f(x) = 2x - 3 \] Thus, \[ f(3) = 2(3) - 3 = 6 - 3 = 3 \]
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Determining which function has a steeper slope:
- The slope of \( f(x) = 2x - 3 \) is 2.
- The slope of \( g(x) = -4x - 1 \) is -4 (considered in absolute value, it's 4). Since the absolute value of -4 is greater than 2, function \( g(x) \) has a steeper slope.
Summary of the first set:
- \( f(3) = 3 \)
- The function with the steeper slope is \( g(x) \).
Second set of functions:
- Finding the y-intercept of each function:
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For \( f(x) = 2x - 3 \): The y-intercept is when \( x = 0 \): \[ f(0) = 2(0) - 3 = -3 \]
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For the equation \( 4x - 7y = 28 \), we need to solve for \( y \): \[ 4x - 7y = 28 \implies -7y = -4x + 28 \implies y = \frac{4}{7}x - 4 \] The y-intercept here is when \( x = 0 \): \[ y(0) = \frac{4}{7}(0) - 4 = -4 \]
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Now we can compare the y-intercepts:
- The y-intercept of \( f(x) \) is -3.
- The y-intercept from \( 4x - 7y = 28 \) is -4.
Thus, \( f(x) \) has a greater y-intercept.
- Calculating \( f(-2) \): \[ f(x) = 2x - 3 \] Therefore, \[ f(-2) = 2(-2) - 3 = -4 - 3 = -7 \]
Summary of the second set:
- The function with the greater y-intercept is \( f(x) \).
- \( f(-2) = -7 \).