Problem 82 Show that if $f$ is a nonconst... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 2: Problem 82

Show that if $f$ is a nonconstant linear function and $g$ is a quadratic function, then $f \circ g$ and $g \circ f$ are both quadratic functions.

Short Answer

Expert verified

In summary, if $f(x) = ax + b$ is a nonconstant linear function and $g(x) = cx^2 + dx + e$ is a quadratic function, then their compositions $f(g(x))$ and $g(f(x))$ are also quadratic functions: 1. $f(g(x)) = acx^2 + adx + ae + b$ 2. $g(f(x)) = ca^2x^2 + (2cab + ad)x + (cb^2 + bd + e)$ These expressions indeed have the quadratic form $kx^2 + lx + m$, with constant coefficients. Therefore, $f \circ g$ and $g \circ f$ are both quadratic functions.

Step by step solution

Define the linear and quadratic functions

We are given that $f$ is a nonconstant linear function, so it can be written in the form $f(x) = ax + b$, where $a \neq 0$ and $b$ are constants. We are also given that $g$ is a quadratic function, so it can be written in the form $g(x) = cx^2 + dx + e$, where $c$, $d$, and $e$ are constants.

Determine the composition functions

Now, we need to find the compositions $f \circ g$ and $g \circ f$. These are given by: 1. $f(g(x)) = f(cx^2 + dx + e)$ 2. $g(f(x)) = g(ax + b)$

Calculate f(g(x))

Let's compute the composition function $f(g(x))$: $ f(g(x)) = f(cx^2 + dx + e) $ Plugging this into the linear function $f(x)$, we get: $ f(g(x)) = a(cx^2 + dx + e) + b $ Expanding this expression, we get: $ f(g(x)) = acx^2 + adx + ae + b $

Calculate g(f(x))

Now let's compute the composition function $g(f(x))$: $ g(f(x)) = g(ax + b) $ Plugging this into the quadratic function $g(x)$, we get: $ g(f(x)) = c(ax + b)^2 + d(ax + b) + e $ Expanding this expression, we get: $ g(f(x)) = c(a^2x^2 + 2abx + b^2) + d(ax + b) + e $ Which further simplifies to: $ g(f(x)) = ca^2x^2 + (2cab + ad)x + (cb^2 + bd + e) $

Check that both functions are quadratic

Now, we must ensure that both composition functions are indeed quadratic. To be considered quadratic, the function must have the form $kx^2 + lx + m$, where $k$, $l$, and $m$ are constants. Observing the results from steps 3 and 4, we can see that: 1. $f(g(x)) = acx^2 + adx + ae + b$ 2. $g(f(x)) = ca^2x^2 + (2cab + ad)x + (cb^2 + bd + e)$ Both of these expressions clearly have the quadratic form $kx^2 + lx + m$, where their coefficients are constants. Therefore, we have shown that if $f$ is a nonconstant linear function and $g$ is a quadratic function, then $f \circ g$ and $g \circ f$ are both quadratic functions.

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91影视

Show that if \(f\) is a nonconstant linear function and \(g\) is a quadratic function, then \(f \circ g\) and \(g \circ f\) are both quadratic functions.

Short Answer

Step by step solution

Define the linear and quadratic functions

Determine the composition functions

Calculate f(g(x))

Calculate g(f(x))

Check that both functions are quadratic

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