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Chapter 6: Problem 4

Given \(g(x)=3 x^{2}-7\) find (a) \(g(3 t)\) (c) \(g(6 t-4)\) (d) \(g(4 x+9)\)

Short Answer

Expert verified
Question: Evaluate the function \(g(x) = 3x^2 - 7\) at the following points: a) \(g(3t)\) c) \(g(6t-4)\) d) \(g(4x+9)\). Answer: a) The expression for \(g(3t)\) is \(27t^2 - 7\). c) The expression for \(g(6t-4)\) is \(108t^2 - 144t + 41\). d) The expression for \(g(4x+9)\) is \(48x^2 + 216x + 236\).

Step by step solution

01

(a) Find g(3t)

To find \(g(3t)\), replace \(x\) with \(3t\) in the given function: \(g(3t) = 3(3t)^2 - 7\) Now, simplify this expression: \(g(3t) = 3(9t^2) - 7\) \(g(3t) = 27t^2 - 7\) The expression for \(g(3t)\) is \(27t^2 - 7\).
02

(c) Find g(6t-4)

To find \(g(6t-4)\), replace \(x\) with \(6t-4\) in the function \(g(x)\): \(g(6t-4) = 3(6t-4)^2 - 7\) Now, simplify this expression: \(g(6t-4) = 3(36t^2 - 48t + 16) - 7\) \(g(6t-4) = 108t^2 - 144t + 48 - 7\) \(g(6t-4) = 108t^2 - 144t + 41\) The expression for \(g(6t-4)\) is \(108t^2 - 144t + 41\).
03

(d) Find g(4x+9)

To find \(g(4x+9)\), replace \(x\) with \(4x+9\) in the given function: \(g(4x+9) = 3(4x+9)^2 - 7\) Now, simplify this expression: \(g(4x+9) = 3(16x^2 + 72x + 81) - 7\) \(g(4x+9) = 48x^2 + 216x + 243 - 7\) \(g(4x+9) = 48x^2 + 216x + 236\) The expression for \(g(4x+9)\) is \(48x^2 + 216x + 236\).

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Most popular questions from this chapter

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) \(f\left(\frac{t}{\alpha}\right)\)

Plot a graph of the following functions. In each case state the domain and the range of the function. (a) \(f(x)=3 x+2,-2 \leq x \leq 5\) (b) \(g(x)=x^{2}+4,-2 \leq x \leq 3\) (c) \(p(t)=2 t^{2}+8,-2 \leq t \leq 4\) (d) \(f(t)=6-t^{2}, 1 \leq t \leq 5\)

Explain why a one-to-many rule cannot be a function.

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A periodic function has period \(0.01\) seconds. How many times will the pattern in the graph repeat over an interval of 10 seconds?
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