Q:

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 between function f and function g

Accepted Solution

A:
The function rule for function g is g(x) = 3³ˣ ⁻ ² + 4The 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)[tex]\texttt{ }[/tex]Further explanationFunction 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 :Linear Function → f(x) = ax + bQuadratic Function → f(x) = ax² + bx + cTrigonometric Function → f(x) = sin x or f(x) = cos x or f(x) = tan xLogarithmic function → f(x) = ln xPolynomial function → f(x) = axⁿ + bxⁿ⁻¹ + ...[tex]\texttt{ }[/tex]If function f : x → y , then inverse function f⁻¹ : y → xLet us now tackle the problem![tex]\texttt{ }[/tex]Given:[tex]f(x) = 3^{x - 2}[/tex][tex]g(x) = f(3x) + 4[/tex]Asked:g(x) = ?Solution:If [tex]f(x) = 3^{x - 2}[/tex] , then:[tex]h(x) = f(3x)[/tex] → Horizontal Scaling by a scale factor of ¹/₃ (Scaling in the x - direction) from graph f(x)[tex]h(x) = 3^{3x - 2}[/tex][tex]\texttt{ }[/tex][tex]g(x) = h(x) + 4[/tex] → Vertical Translation by 4 units upwards (Translation in the y-direction) from graph h(x)[tex]g(x) = 3^{3x - 2} + 4[/tex][tex]\texttt{ }[/tex]Conclusion:The function rule for function g is [tex]\boxed{ g(x) = 3^{3x - 2} + 4 }[/tex]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)[tex]\texttt{ }[/tex]Learn moreInverse of Function : of Change : of Function : detailsGrade: High SchoolSubject: MathematicsChapter: FunctionKeywords: Function , Trigonometric , Linear , Quadratic, Transformations#LearnWithBrainly