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Chapter 9: Problem 18

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(g(0.5)\)

Short Answer

Expert verified
g(0.5) = 2.75

Step by step solution

01

Identify the Function

Determine which function to evaluate at the given value. In this case, use the function for which you need to find the value, which is \(g(x)\).
02

Substitute the Given Value

Replace the variable \(x\) in \(g(x)\) with the given value 0.5. This gives \(g(0.5) = - (0.5)^{2} + 4(0.5) + 1\).
03

Evaluate the Expression

Calculate each term separately: \((0.5)^{2} = 0.25\), \(4 \times 0.5 = 2\). Plugging these into the equation: \(g(0.5) = -0.25 + 2 + 1\).
04

Simplify the Result

Combine the terms to get the final value: \(g(0.5) = 2 + 1 - 0.25 = 2.75\).

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Key Concepts

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

Substitution
Substitution is a fundamental concept in mathematics. It involves replacing a variable in an expression with a given value. This process helps in simplifying and evaluating functions or equations. For example, in our exercise, we substitute the value 0.5 into the function \(g(x)\). This method is useful for checking solutions or evaluating given functions at specific points. Remember to always replace every occurrence of the variable with the value provided.
Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. These functions have the general form: \[P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where each term is a power of \(x\) multiplied by a coefficient. In our exercise, \(g(x) = -x^2 + 4x + 1\) is a polynomial function. Understanding the structure of polynomial functions helps in performing operations like differentiation, integration, and function evaluation. They appear frequently in various fields like physics, economics, and engineering.
Evaluating Expressions
Evaluating expressions means finding the value of an expression after substituting the variables with specific numbers. This involves performing arithmetic operations following the order of operations (PEMDAS/BODMAS). In the given problem, after substituting \(x\) with 0.5 in \(g(x)\), we perform each arithmetic operation step by step: calculating the square, multiplying by coefficients, combining terms, and simplifying. Step-by-step evaluation ensures accuracy and helps in understanding the behavior of mathematical functions.

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Most popular questions from this chapter

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ g)(4) $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ h)(0) $$

The perimeter \(x\) of a square with sides of length \(s\) is given by the formula \(x=4 s\) (a) Solve for \(s\) in terms of \(x\). (b) If \(y\) represents the area of this square, write \(y\) as a function of the perimeter \(x\). (c) Use the composite function of part (b) to find the area of a square with perimeter 6 .

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=-x $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following $$ (g h)(-2) $$
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