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Chapter 0: Problem 20

Evaluate the indicated function for \(f(x)=x^{2}+1\) and \(g(x)=x-4\) $$(f g)(-6)$$

Short Answer

Expert verified
The value of the function composition (f g)(-6) is 101.

Step by step solution

01

Find g(-6)

To begin, we find the value of g at -6. So, substitute \(x = -6\) into \(g(x) = x - 4\). Hence, \(g(-6) = -6 - 4 = -10\).
02

Find (f g)(-6)

Next, substitute \(g(-6) = -10\) into the function \(f(x)\). Thus, \(f(g(-6)) = f(-10)\). Now, substitute \(x = -10\) into \(f(x) = x^{2} + 1\). Therefore, \(f(-10) = (-10)^{2} + 1 = 100 + 1 = 101\).
03

Result

So, the value of the function composition (f g)(-6) is 101.

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Key Concepts

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

Evaluate Composite Functions
Composite functions involve evaluating multiple functions in succession. They are important in mathematics as they allow us to cascade processes. For example, consider two functions, \( f(x) \) and \( g(x) \). A composite function \((f \circ g)(x)\) is evaluated by first applying \( g \), then applying \( f \) on the result from \( g \).
  • Step 1: First, calculate the inner function \( g(x) \). For the given exercise, this involves substituting a specific value into \( g(x) = x - 4 \).
  • Step 2: Use the result from \( g(x) \) as the input for \( f(x) \). Thus, substitute back into \( f(x) = x^2 + 1 \) to get the final answer.
The order of operations is critical; always solve the innermost function first. This concept is handy in scenarios where functions describe real-world processes, such as converting coordinates or making calculations in engineering.
Quadratic Functions
Quadratic functions, like \( f(x) = x^2 + 1 \), are polynomial functions of degree 2. They produce a parabolic graph that opens upwards if the coefficient of \( x^2 \) is positive.
  • The standard form of a quadratic function is \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants.
  • The vertex form is another useful representation, given by \( a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
  • They have important properties such as vertex, axis of symmetry, and roots, which are the points where the parabola intersects the x-axis.
In this exercise, substituting into the quadratic function involves squaring the input, then adjusting by adding one. This showcases how quadratic functions can transform values effectively.
Linear Functions
Linear functions are the simplest form of polynomial functions and have a constant rate of change. In this exercise, the function \( g(x) = x - 4 \) is a linear function.
  • They are of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • The graph of a linear function is a straight line.
  • The slope \( m \) represents the change in \( y \) for a unit change in \( x \).
For \( g(x) = x - 4 \), the slope is \( 1 \), and it shifts the graph downward by 4 units. This simple transformation highlights the straightforward nature of linear functions.

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Most popular questions from this chapter

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

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt{x} $$

(a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

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?

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 120 feet. $$ t_{1}=0, t_{2}=2 $$
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