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Chapter 0: Problem 18

Evaluate the indicated function for \(f(x)=x^{2}+1\) and \(g(x)=x-4\) $$(f+g)(t-2)$$

Short Answer

Expert verified
The value of the function \((f+g)(t-2)\) is \(t^{2}-3t-1\).

Step by step solution

01

Understand Function Addition

Function addition refers to adding the outputs of two separate functions. It is denoted as (f+g)(x). In this case, the functions are \(f(x)=x^{2}+1\) and \(g(x)=x-4\). When we add these two functions, we get \((f+g)(x) = f(x) + g(x) = (x^{2}+1) + (x-4)\). So, \((f+g)(x)=x^{2}+x-3\).
02

Substitute the Expression

Now, we substitute \(t-2\) for \(x\) in the combined function. So \((f+g)(t-2) = (t-2)^{2}+(t-2)-3\).
03

Simplify the Expression

Simplifying this expression gives \((f+g)(t-2) = t^{2}-4t+4+t-2-3 = t^{2}-3t-1\).

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Key Concepts

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

Function Addition
Function addition is a basic operation that combines two functions by summing their output values. For two functions, \(f(x)\) and \(g(x)\), the sum is represented as \((f+g)(x)\). This simply means adding each function鈥檚 result for a given input \(x\).
For example, given \(f(x) = x^2 + 1\) and \(g(x) = x - 4\), we find their sum:
  • First, evaluate \(f(x) = x^2 + 1\).
  • Then, evaluate \(g(x) = x - 4\).
  • Add the outputs: \(f(x) + g(x) = (x^2 + 1) + (x - 4)\).
After simplifying, \((f+g)(x) = x^2 + x - 3\). This is the combined or resulting function.
Polynomial Functions
Polynomial functions are expressions constructed from variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. They are a cornerstone of algebra and play a crucial role in calculus and various areas of mathematics.
One of the simplest forms of polynomial function is the quadratic polynomial, like \(f(x) = x^2 + 1\). In our context, \(f(x) = x^2+1\) and \(g(x) = x-4\) are both polynomials.
Let's break down their features:
  • \(f(x)\) is a quadratic polynomial because its highest exponent is 2.
  • \(g(x)\) is a linear polynomial because its highest exponent is 1.
Polynomial functions like these form a sequence of terms that define their behavior graphically and numerically.
Substitution and Simplification
Substitution is a method used to replace a variable in an expression with a given value or another expression. It's a key technique in evaluating functions at specific points.
In the given problem, after forming the function \((f+g)(x) = x^2 + x - 3\), we substitute \(x\) with \(t-2\). This transforms the function to \((f+g)(t-2) = (t-2)^2 + (t-2) - 3\).
Next, we simplify this expression through arithmetic operations:
  • Expand the squared term: \((t-2)^2 = t^2 - 4t + 4\).
  • Add \(t-2\) to get \(t^2 - 4t + 4 + t - 2\).
  • Combine like terms: \(t^2 - 3t + 2\).
  • Subtract 3 to finish simplification: \(t^2 - 3t - 1\).
This final expression, \(t^2 - 3t - 1\), is the result of substituting and simplifying the given polynomial function.

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Most popular questions from this chapter

The numbers of households \(f\) (in thousands) in the United States from 1995 to 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Households, } f(t) \\ \hline 5 & 98,990 \\ 6 & 99,627 \\ 7 & 101,018 \\ 8 & 102,528 \\ 9 & 103,874 \\ 10 & 104,705 \\ 11 & 108,209 \\ 12 & 109,297 \\ 13 & 111,278 \\ \hline \end{array} $$ (a) Find \(f^{-1}(108,209)\). (b) What does \(f^{-1}\) mean in the context of the problem? (c) Use the regression feature of a graphing utility to find a linear model for the data, \(y=m x+b\). (Round \(m\) and \(b\) to two decimal places.) (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(117,022)\). (f) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(108,209)\). How does this value compare with the original data shown in the table?

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=x^{4} $$

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ g^{-1} \cdot f^{-1} $$

Average Price The average prices \(p\) (in thousands of dollars) of a new mobile home in the United States from 1990 to 2002 (see figure) can be approximated by the model $$ p(t)= \begin{cases}0.182 t^{2}+0.57 t+27.3, & 0 \leq t \leq 7 \\ 2.50 t+21.3, & 8 \leq t \leq 12\end{cases} $$ where \(t\) represents the year, with \(t=0\) corresponding to 1990. Use this model to find the average price of a mobile home in each year from 1990 to 2002 . (Source: U.S. Census Bureau)

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\sqrt{2 x+3} $$
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