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Answer the questions below to determine what kind of function is depicted in the table below.

xx 11 22 33 44 55
f, of, xf(x) 22 1313 4646 101101 178178
Answer

This function is quadratic, linear, exponential or none of the above?
because?

3 answers

To determine the type of function depicted in the table, let's examine the relationship between the input values \( x \) and the output values \( f(x) \).

Given the input values:

  • \( x = 1, f(1) = 22 \)
  • \( x = 2, f(2) = 1313 \)
  • \( x = 3, f(3) = 4646 \)
  • \( x = 4, f(4) = 101101 \)
  • \( x = 5, f(5) = 178178 \)

Step 1: Check for linearity

A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Let's check the first differences (the change in \( f(x) \)):

  • \( f(2) - f(1) = 1313 - 22 = 1291 \)
  • \( f(3) - f(2) = 4646 - 1313 = 3333 \)
  • \( f(4) - f(3) = 101101 - 4646 = 96655 \)
  • \( f(5) - f(4) = 178178 - 101101 = 77077 \)

The first differences (1291, 3333, 96655, 77077) are not constant, so the function is not linear.

Step 2: Check for quadratic nature

A quadratic function can be expressed in the form \( f(x) = ax^2 + bx + c \). To check if the function might be quadratic, we can calculate the second differences (the differences of the first differences):

Calculate the second differences:

  • \( 3333 - 1291 = 2042 \)
  • \( 96655 - 3333 = 93322 \)
  • \( 77077 - 96655 = -19578 \)

The second differences (2042, 93322, -19578) are not constant, so the function is not quadratic.

Step 3: Check for exponential nature

An exponential function can be expressed as \( f(x) = a \cdot b^x \). We would check if the ratio of successive outputs is constant:

Calculate the ratios:

  • \( \frac{f(2)}{f(1)} = \frac{1313}{22} \approx 59.636 \)
  • \( \frac{f(3)}{f(2)} = \frac{4646}{1313} \approx 3.54 \)
  • \( \frac{f(4)}{f(3)} = \frac{101101}{4646} \approx 21.73 \)
  • \( \frac{f(5)}{f(4)} = \frac{178178}{101101} \approx 1.76 \)

The ratios are not constant, hence, the function is not exponential.

Conclusion

Since the function is neither linear, quadratic, nor exponential, we conclude that the function is none of the above.

why tho

Let’s break down why the function is classified as “none of the above” when examining the relationships in the given table:

1. Linearity

A linear function has the form \( f(x) = mx + b \). This means that if you were to plot the function, the result would be a straight line. For a linear function:

  • The first differences (the change in \( f(x) \) values) should be constant.

Calculating the first differences:

  • \( f(2) - f(1) = 1313 - 22 = 1291 \)
  • \( f(3) - f(2) = 4646 - 1313 = 3333 \)
  • \( f(4) - f(3) = 101101 - 4646 = 96655 \)
  • \( f(5) - f(4) = 178178 - 101101 = 77077 \)

Since these first differences (1291, 3333, 96655, 77077) are not constant, the function is not linear.

2. Quadratic

A quadratic function has a form of \( f(x) = ax^2 + bx + c \). For such a function:

  • The first differences between the outputs will not be constant, but the second differences (the differences of the first differences) will be constant.

From earlier: First differences: \( 1291, 3333, 96655, 77077 \)

Calculating second differences:

  • \( 3333 - 1291 = 2042 \)
  • \( 96655 - 3333 = 93322 \)
  • \( 77077 - 96655 = -19578 \)

Since these second differences (2042, 93322, -19578) are not constant, the function is not quadratic.

3. Exponential

An exponential function generally takes the form \( f(x) = a \cdot b^x \) where, as \( x \) increases, \( f(x) \) grows at a rate proportional to its current value.

  • For an exponential function, the ratios of successive outputs should be constant.

Calculating the ratios:

  • \( \frac{f(2)}{f(1)} = \frac{1313}{22} \approx 59.636 \)
  • \( \frac{f(3)}{f(2)} = \frac{4646}{1313} \approx 3.54 \)
  • \( \frac{f(4)}{f(3)} = \frac{101101}{4646} \approx 21.73 \)
  • \( \frac{f(5)}{f(4)} = \frac{178178}{101101} \approx 1.76 \)

Since these ratios (approximately 59.636, 3.54, 21.73, 1.76) are not constant, the function is not exponential.

Conclusion

Since the function does not fit the characteristics of linear, quadratic, or exponential forms, we conclude that it is classified as none of the above.

The patterns of growth and the specific outputs do not align with standard mathematical function types, indicating that it may follow a different, more complex rule or is simply an irregular relationship not captured by these common classifications.