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Chapter 6: Problem 3

Given the function \(f(t)=2 t^{2}+4\) find (a) \(f(x)\) (b) \(f(2 x)\) (c) \(f(-x)\) (d) \(f(4 x+2)\) (e) \(f(3 t+5)(\mathrm{f}) f(\lambda)(\mathrm{g}) f(t-\lambda)\) (h) \(f\left(\frac{t}{\alpha}\right)\)

Short Answer

Expert verified
Answer: f(4x + 2) = 32x^2 + 32x + 12

Step by step solution

01

Replace t with x

Substitute \(t\) with \(x\) in the function \(f(t)\). \(f(x) = 2x^2 + 4\) This is the function \(f\) where the variable is changed to \(x\). #b) f(2x)#
02

Replace t with 2x

Substitute \(t\) with \(2x\) in the function \(f(t)\). \(f(2x) = 2(2x)^2 + 4\) Now, simplify the expression: \(f(2x) = 2(4x^2) + 4 = 8x^2 + 4\) #c) f(-x)#
03

Replace t with -x

Substitute \(t\) with \(-x\) in the function \(f(t)\). \(f(-x) = 2(-x)^2 + 4\) Now, simplify the expression: \(f(-x) = 2(x^2) + 4 = 2x^2 + 4\) #d) f(4x + 2)#
04

Replace t with 4x + 2

Substitute \(t\) with \(4x + 2\) in the function \(f(t)\). \(f(4x + 2) = 2(4x + 2)^2 + 4\) Now, simplify the expression: \(f(4x + 2) = 2(16x^2 + 16x + 4) + 4 = 32x^2 + 32x + 12\) #e) f(3t + 5)#
05

Replace t with 3t + 5

Substitute \(t\) with \(3t + 5\) in the function \(f(t)\). \(f(3t + 5) = 2(3t + 5)^2 + 4\) Now, simplify the expression: \(f(3t + 5) = 2(9t^2 + 30t + 25) + 4 = 18t^2 + 60t + 54\) #f) f(λ)#
06

Replace t with λ

Substitute \(t\) with \(λ\) in the function \(f(t)\). \(f(λ) = 2λ^2 + 4\) This is the function \(f\) where the variable is changed to \(λ\). #g) f(t - λ)#
07

Replace t with t - λ

Substitute \(t\) with \(t - λ\) in the function \(f(t)\). \(f(t - λ) = 2(t - λ)^2 + 4\) Now, simplify the expression: \(f(t - λ) = 2(t^2 - 2tλ + λ^2) + 4 = 2t^2 - 4tλ + 2λ^2 + 4\) #h) f(t/α)#
08

Replace t with t/α

Substitute \(t\) with \(t/α\) in the function \(f(t)\). \(f\left(\frac{t}{α}\right) = 2\left(\frac{t}{α}\right)^2 + 4\) Now, simplify the expression: \(f\left(\frac{t}{α}\right) = 2\left(\frac{t^2}{α^2}\right) + 4 = \frac{2t^2}{α^2} + 4\)

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If \(f(x)=x+6\) and \(g(x)=x^{2}-5\) find (a) \(f(g(0))\), (b) \(g(f(0))\), (c) \(g(g(2))\), (d) \(f(g(7))\).

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