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Chapter 2: Problem 28

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(t) $$

Short Answer

Expert verified
g(t) = -t^2 + 4t + 1

Step by step solution

01

Identify the function for g(x)

The function given for g(x) is \[g(x) = -x^2 + 4x + 1.\]
02

Substitute t for x in g(x)

To find \( g(t) \), replace every instance of \( x \) in \( g(x) \) with \( t \). This gives: \[ g(t) = -t^2 + 4t + 1. \]
03

Simplify if necessary

Check to see if the new function \( g(t) = -t^2 + 4t + 1 \) can be simplified further. In this case, it is already in its simplest form.

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Key Concepts

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

substitution in functions
A powerful tool in mathematics is substitution in functions. Substitution involves replacing one variable with another to evaluate the function for different values. For example, if you have a function such as \( g(x) = -x^2 + 4x + 1 \), and you need to find \(g(t)\), you will replace every instance of the variable \( x \) with \( t \). This gives you the new function \( g(t) = -t^2 + 4t + 1 \). Substitution is straightforward, but it requires careful attention to ensure every occurrence of the variable is replaced correctly. It's also a technique used in calculus and algebra for solving equations.
polynomial functions
Polynomial functions are expressions characterized by variables raised to whole number exponents and coefficients. These functions are a fundamental part of algebra and can take various forms. For instance, the polynomial function \(g(x)\) given in the exercise is \[g(x) = -x^2 + 4x + 1.\]Here,
  • The highest power of \(x\) is 2, which means \(g(x)\) is a quadratic polynomial.
  • The coefficients are -1 for \( x^2 \), 4 for \(x\), and 1 is the constant term.
Polynomial functions can represent many real-world scenarios, such as the trajectory of a ball or various economic models. Understanding their structure helps you simplify and manipulate them for solving complex problems.
simplifying expressions
Simplifying expressions means reducing them to their simplest form. This makes them easier to work with and solve. For polynomial functions like the one we have \[g(t) = -t^2 + 4t + 1,\]ensure that there are no further possible reductions by combining like terms or factoring. In this instance, \(g(t)\) is already in its simplest form. However, for more complicated expressions, follow these steps:
  • Combine like terms. These are terms that have the same variable and exponent.
  • Look for and remove common factors.
  • Apply algebraic identities where applicable, like the difference of squares or the distributive property.
Mastering simplification is crucial for efficiently solving equations and understanding higher-level mathematics.

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Most popular questions from this chapter

The upper deck at Guaranteed Rate Field in Chicago has produced, among other complaints, displeasure with its steepness. It is \(160 \mathrm{ft}\) from home plate to the front of the upper deck and \(250 \mathrm{ft}\) from home plate to the back. The top of the upper deck is \(63 \mathrm{ft}\) above the bottom. What is its slope?

The average price of a movie ticket in 2004 was \(\$ 6.21 .\) In \(2016,\) the average price was \(\$ 8.65 .\) Find and interpret the average rate of change in the price of a movie ticket per year to the nearest cent.

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ g(\pi) $$

Graph each line passing through the given point and having the given slope. (5,3)\(; m=0\)

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ 4 x-3 y=8 $$
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