given f(x)=3^x-2 and g(x)=f(3x)+4, write the function rule for function g and describe the types of transformations that occur b

Question

jek_recluse [69]

3 years ago

10

given f(x)=3^x-2 and g(x)=f(3x)+4, write the function rule for function g and describe the types of transformations that occur b

etween function f and function g

Mathematics

2 answers:

Dmitriy789 [7] · Answer 1 · 2022-07-30T15:01:07+03:00

The function rule for function g is g(x) = 3³ˣ ⁻ ² + 4

The types of transformations that occur between function f and function g are horizontal scaling by a scale factor of ¹/₃ and followed by vertical translation by 4 units upwards from graph f(x)

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<h3>Further explanation</h3>

Function is a relation which each member of the domain is mapped onto exactly one member of the codomain.

There are many types of functions in mathematics such as :

<em>Linear Function → f(x) = ax + b</em>

<em>Quadratic Function → f(x) = ax² + bx + c</em>
<em>Trigonometric Function → f(x) = sin x or f(x) = cos x or f(x) = tan x</em>
<em>Logarithmic function → f(x) = ln x</em>
<em>Polynomial function → f(x) = axⁿ + bxⁿ⁻¹ + ...</em>

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If function f : x → y , then inverse function f⁻¹ : y → x

Let us now tackle the problem!

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<u>Given:</u>

$f(x) = 3^{x - 2}$

$g(x) = f(3x) + 4$

<u>Asked:</u>

g(x) = ?

<u>Solution:</u>

If $f(x) = 3^{x - 2}$ , then:

$h(x) = f(3x)$ → Horizontal Scaling by a scale factor of ¹/₃ (Scaling in the x - direction) from graph f(x)

$h(x) = 3^{3x - 2}$

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$g(x) = h(x) + 4$ → Vertical Translation by 4 units upwards (Translation in the y-direction) from graph h(x)

$g(x) = 3^{3x - 2} + 4$

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<h3>Conclusion:</h3>

The function rule for function g is $\boxed{ g(x) = 3^{3x - 2} + 4 }$

The types of transformations that occur between function f and function g are horizontal scaling by a scale factor of ¹/₃ and followed by vertical translation by 4 units upwards from graph f(x)

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<h3>Learn more</h3>