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2) The logarithmic function [tex]\( y = \log x \)[/tex] is transformed in three different ways. Based on each description given, fill in the blank with a possible equation of the transformed function (with [tex]\( y = \log x \)[/tex] as the original function each time) and justify why your equation corresponds. [6 marks]

a) Equation A: [tex]\[\qquad\][/tex] Justification:

A single transformation is applied to the original function, and the point [tex]\((10, 1)\)[/tex] becomes the point [tex]\((10, -3)\)[/tex] on the transformed function.

b) Equation B: [tex]\[\qquad\][/tex] Justification:

Transformed domain: [tex]\(\{x \in \mathbb{R} \mid x \ \textless \ 4\}\)[/tex]
Transformed range: [tex]\(\{y \in \mathbb{R}\}\)[/tex]

c) Equation C: [tex]\[\qquad\][/tex] Justification:

Mapping notation: [tex]\((x, y) \rightarrow \left(\frac{1}{2} x, -3 y\right)\)[/tex]

Sagot :

GinnyAnswer
Let's delve into each transformation step-by-step, using the original function [tex]\( y = \log x \)[/tex]:

### a) Equation A:
Equation: [tex]\( y = -3 \log x \)[/tex]

Justification:
For the original function [tex]\( y = \log x \)[/tex], the point [tex]\((10, 1)\)[/tex] means that [tex]\(\log 10 = 1\)[/tex]. The transformation applied here takes the y-coordinate of the point and multiplies it by [tex]\(-3\)[/tex]. Hence, the new point is [tex]\((10, -3)\)[/tex].

Thus, the transformation applied to the original function [tex]\( y = \log x \)[/tex] is simply multiplying the entire function by [tex]\(-3\)[/tex]. This gives us:
[tex]\[ y = -3 \log x \][/tex]

### b) Equation B:
Equation: [tex]\( y = \log (4 - x) \)[/tex]

Justification:
The domain of the original function [tex]\( y = \log x \)[/tex] is [tex]\( x > 0 \)[/tex]. For the transformed function, the domain is given as [tex]\( x < 4 \)[/tex]. To achieve this, we need to shift the function horizontally to the right by 4 units and reflect it over the y-axis.

The horizontal shift of 4 units to the right for [tex]\( y = \log x \)[/tex] would normally be represented as [tex]\( y = \log (x - 4) \)[/tex]. To reflect the x-values and achieve the domain requirement [tex]\( x < 4 \)[/tex], we use [tex]\( y = \log (4 - x) \)[/tex].

So, the transformed equation is:
[tex]\[ y = \log (4 - x) \][/tex]

### c) Equation C:
Equation: [tex]\( y = -3 \log (2x) \)[/tex]

Justification:
The mapping notation [tex]\((x, y) \rightarrow \left(\frac{1}{2}x, -3y\right)\)[/tex] indicates two transformations:
- The x-coordinate is scaled by [tex]\(\frac{1}{2}\)[/tex], meaning each input x is transformed to [tex]\(\frac{x}{2}\)[/tex].
- The y-coordinate is scaled by [tex]\(-3\)[/tex], implying the output is multiplied by [tex]\(-3\)[/tex].

Firstly, for the input [tex]\( x \rightarrow 2x \)[/tex], it means every x value in the original function should be modified to [tex]\(2x\)[/tex] (since [tex]\(\frac{1}{2}x\)[/tex] reversed would be [tex]\(2x\)[/tex]).

Secondly, resulting y values are scaled by [tex]\(-3\)[/tex], hence:
[tex]\[ y = -3 \log (2x) \][/tex]

This results in the final transformed equation as:
[tex]\[ y = -3 \log (2x) \][/tex]
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