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Chapter 1: Problem 21

Evaluate the indicated function, where \(f(x)=x^{2}-3 x+2\) and \(g(x)=2 x-4\). $$(f g)(7)$$

Short Answer

Expert verified
The result of the composition of the function at x=7 is 72.

Step by step solution

01

Substitute function g into function f

We have to substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). This results in \(f(x)= (2x - 4)^{2}-3(2x - 4)+2\).
02

Evaluate the function at x=7

Now we substitute \(x=7\) into the function, which results in \((f \circ g)(7) = (2*7 - 4)^{2}-3(2*7 - 4)+2\).
03

Carry out the arithmetic operations

Carrying out the arithmetic operations, we get \( (f \circ g)(7)= (14 - 4)^{2}-3(14 - 4)+2 = 10^{2} - 30 + 2 = 100 - 30 + 2 = 72\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polynomial Functions
A polynomial function is an expression made up of variables and constants, combined using only addition, subtraction, multiplication, and non-negative integer exponents. Essentially, polynomials are combinations of terms structured like sums of powers of a variable.
For example, in the function \(f(x) = x^2 - 3x + 2\), we have three terms: \(x^2\), \(-3x\), and \(+2\).
  • The term \(x^2\) is a second-degree polynomial, also known as a quadratic term.
  • The term \(-3x\) is a first-degree term or linear term.
  • Finally, \(+2\) is a constant term.
These terms are combined to form the polynomial function. A key aspect of polynomial functions is their smooth graphs, often taking the shape of a gentle curve rather than sharp turns or corners.
Exploring Function Evaluation
Function evaluation is the process of finding the value of a function for a specific input. This is a crucial aspect of working with functions.
When evaluating a function, you're essentially replacing the variable in the function's formula with a given number.
In our exercise, we first evaluated the composite function \(f(g(x))\) at \(x = 7\). This means we calculated the result of the function \(f(x)\) when \(g(x)\) takes the place of \(x\).
  • For example, substituting \(7\) into \(g(x) = 2x - 4\), we computed \(g(7) = 2 \times 7 - 4\).
  • The value \(g(7) = 10\) was then plugged into \(f(x)\).
This process of taking one function's output and using it as another function's input is the essence of function composition, offering a powerful way to construct complex expressions.
Performing Arithmetic Operations
Arithmetic operations on functions involve the basic operations: addition, subtraction, multiplication, and, in some cases, division.
In our problem, we focused on substitution and multiplication of terms鈥攌ey operations for evaluating the composite function \((f \circ g)(7)\).
  • At first, it involved squaring the expression \((14 - 4)\).
  • Next up was the subtraction: \(-3 \times (14 - 4)\).
  • Lastly, we added the constant \(+2\).
We combined these operations to get the final result.
The steps of performing these arithmetic operations methodically are crucial for accuracy, ensuring we reach the correct value for the input.

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Most popular questions from this chapter

Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$-4(x-5)=2 x+15$$

Use interval notation to express the solution set of each inequality. $$|3 x-10| \leq 14$$

Use interval notation to express the solution set of each inequality. $$|x-4| \leq 0$$

Given the points \(P_{1}(-3,4)\) and \(P_{2}(2,-4),\) evaluate \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$-5 x+2 \leq 37$$
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