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Given the function f(x) = 4x − 1.

Computer Science

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Answer

a
10.996
11.004
Graph of f(x) = 4x − 1

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Key points of f(x) = 4x − 1
Pointxy
y-intercept0−1
x-intercept0.250

Step-by-step solution

2 steps

Here's the solution to the problem:

Given the function f(x)=4x−1f(x) = 4x - 1.

Part a· 2 steps

Find the value of f(2.999)f(2.999) and f(3.001)f(3.001).

  1. Calculate f(2.999)f(2.999).

    Substitute x=2.999x = 2.999 into the function:

    f(2.999)=4(2.999)−1f(2.999) = 4(2.999) - 1 f(2.999)=11.996−1f(2.999) = 11.996 - 1 f(2.999)=10.996f(2.999) = 10.996
  2. Calculate f(3.001)f(3.001).

    Substitute x=3.001x = 3.001 into the function:

    f(3.001)=4(3.001)−1f(3.001) = 4(3.001) - 1 f(3.001)=12.004−1f(3.001) = 12.004 - 1 f(3.001)=11.004f(3.001) = 11.004

    The values are f(2.999)=10.996f(2.999) = \boxed{10.996} and f(3.001)=11.004f(3.001) = \boxed{11.004}.

Part b

Is function f(x)f(x) continuous at the point x=3x=3? Give reason.

Yes, the function f(x)=4x−1f(x) = 4x - 1 is continuous at the point x=3x=3.

Reason:
A function f(x)f(x) is continuous at a point x=cx=c if three conditions are met:

  1. f(c)f(c) is defined.
  2. lim⁡x→cf(x)\lim_{x \to c} f(x) exists.
  3. lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c).

For f(x)=4x−1f(x) = 4x - 1 at x=3x=3:

  1. f(3)=4(3)−1=12−1=11f(3) = 4(3) - 1 = 12 - 1 = 11. So, f(3)f(3) is defined.
  2. lim⁡x→3(4x−1)=4(3)−1=11\lim_{x \to 3} (4x - 1) = 4(3) - 1 = 11. So, the limit exists.
  3. lim⁡x→3f(x)=11\lim_{x \to 3} f(x) = 11 and f(3)=11f(3) = 11. Thus, lim⁡x→3f(x)=f(3)\lim_{x \to 3} f(x) = f(3).

Alternatively, and more concisely, f(x)=4x−1f(x) = 4x - 1 is a polynomial function. All polynomial functions are continuous for all real numbers. Therefore, f(x)f(x) is continuous at x=3x=3.

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