Verified Answer :
- For each function, we systematically substituted the input values \(x = -2, 0,\) and \(5\).
- Using basic algebraic principles and order of operations, we calculated the output for every pair.
- The results are summarized in a table showing the outputs for \(f(x), g(x), h(x), r(x), p(x),\) and \(b(x)\) at each input value.
- Final outputs at \(x = \{-2, 0, 5\}\):
\(f(x): \{4, 6, 11\}\)
\(g(x): \{-6, 0, 15\}\)
\(h(x): \{13, 9, -1\}\)
\(r(x): \{-5, -7, -12\}\)
\(p_1(x): \{-3.5, -3, -1.75\}\)
\(b_1(x): \{19, 18, 15.5\}\)
\(p_2(x): \{5, -3, 17\}\)
\(b_2(x): \{20, 18, 30.5\}\)
Explanation
- Problem Analysis
To evaluate these functions, we substitute each value of \(x\) (-2, 0, and 5) into the given algebraic expressions. This process involves basic arithmetic, including addition, subtraction, multiplication, division, exponentiation, and the application of the absolute value function.
The functions are:
- \(f(x) = x + 6\)
- \(g(x) = 3x\)
- \(h(x) = -2x + 9\)
- \(r(x) = -x - 7\)
- \(p(x) = -3 + \frac{1}{4}x\)
- \(b(x) = 18 - 0.5x\)
- \(p(x) = -3 + |4x|\)
- \(b(x) = 18 + 0.5x^2\)
- Step-by-Step Evaluation
We calculate the values for each function at the specified inputs of \(x = -2, 0, 5\):
For \(x = -2\):
- \(f(-2) = -2 + 6 = 4\)
- \(g(-2) = 3(-2) = -6\)
- \(h(-2) = -2(-2) + 9 = 4 + 9 = 13\)
- \(r(-2) = -(-2) - 7 = 2 - 7 = -5\)
- \(p(-2) = -3 + \frac{1}{4}(-2) = -3 - 0.5 = -3.5\)
- \(b(-2) = 18 - 0.5(-2) = 18 + 1 = 19.0\)
- \(p(-2) = -3 + |4(-2)| = -3 + |-8| = -3 + 8 = 5\)
- \(b(-2) = 18 + 0.5(-2)^2 = 18 + 0.5(4) = 18 + 2 = 20.0\)
For \(x = 0\):
- \(f(0) = 0 + 6 = 6\)
- \(g(0) = 3(0) = 0\)
- \(h(0) = -2(0) + 9 = 9\)
- \(r(0) = -0 - 7 = -7\)
- \(p(0) = -3 + \frac{1}{4}(0) = -3 + 0 = -3.0\)
- \(b(0) = 18 - 0.5(0) = 18 - 0 = 18.0\)
- \(p(0) = -3 + |4(0)| = -3 + 0 = -3\)
- \(b(0) = 18 + 0.5(0)^2 = 18 + 0 = 18.0\)
For \(x = 5\):
- \(f(5) = 5 + 6 = 11\)
- \(g(5) = 3(5) = 15\)
- \(h(5) = -2(5) + 9 = -10 + 9 = -1\)
- \(r(5) = -5 - 7 = -12\)
- \(p(5) = -3 + \frac{1}{4}(5) = -3 + 1.25 = -1.75\)
- \(b(5) = 18 - 0.5(5) = 18 - 2.5 = 15.5\)
- \(p(5) = -3 + |4(5)| = -3 + |20| = -3 + 20 = 17\)
- \(b(5) = 18 + 0.5(5)^2 = 18 + 0.5(25) = 18 + 12.5 = 30.5\)
- Final Results and Summary
The results for the functions at each input are summarized as follows:
| Function | \(x=-2\) | \(x=0\) | \(x=5\) |
| :--- | :--- | :--- | :--- |
| \(f(x)=x+6\) | 4 | 6 | 11 |
| \(g(x)=3x\) | -6 | 0 | 15 |
| \(h(x)=-2x+9\) | 13 | 9 | -1 |
| \(r(x)=-x-7\) | -5 | -7 | -12 |
| \(p(x)=-3+0.25x\) | -3.5 | -3 | -1.75 |
| \(b(x)=18-0.5x\) | 19 | 18 | 15.5 |
| \(p(x)=-3+|4x|\) | 5 | -3 | 17 |
| \(b(x)=18+0.5x^2\) | 20 | 18 | 30.5 |
These evaluations complete the requirement to find the outputs for each function given the inputs.
Examples
Evaluating functions at specific points is fundamental in many real-world fields, such as engineering and economics. For example, if a company models its daily revenue \(R\) as a function of the number of items sold \(x\), they might use a formula like \(R(x) = 20x - 0.5x^2\). By calculating \(R(10)\), \(R(20)\), and \(R(30)\), managers can predict how revenue changes at different production levels and make data-driven decisions to maximize profits. Understanding how to evaluate functions allows you to interpret these mathematical models and apply them to solve practical problems effectively.