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If the range of [tex]$f(x)=\sqrt{m x}$[/tex] and the range of [tex]$g(x)=m \sqrt{x}$[/tex] are the same, which statement is true about the value of [tex]m[/tex]?

A. [tex]m[/tex] can only equal 1.
B. [tex]m[/tex] can be any positive real number.
C. [tex]m[/tex] can be any negative real number.
D. [tex]m[/tex] can be any real number.

Sagot :

GinnyAnswer
Sure, let's analyze the problem step-by-step:

Given:
1. [tex]\( f(x) = \sqrt{mx} \)[/tex]
2. [tex]\( g(x) = m\sqrt{x} \)[/tex]

We need to identify the conditions under which the ranges (outputs) of both functions are the same and interpret what this implies about the value of [tex]\( m \)[/tex].

### Step 1: Analyze the range of [tex]\( f(x) = \sqrt{mx} \)[/tex]
- For [tex]\( f(x) \)[/tex] to be defined, the expression inside the square root must be non-negative: [tex]\( mx \ge 0 \)[/tex].
- Since [tex]\( x \ge 0 \)[/tex] for the square root function to be defined in real numbers:
- If [tex]\( m > 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] will yield non-negative values, i.e., any non-negative real number as [tex]\( x \)[/tex] ranges over all non-negative values.

### Step 2: Analyze the range of [tex]\( g(x) = m\sqrt{x} \)[/tex]
- For [tex]\( g(x) \)[/tex] to be defined, the expression inside the square root (i.e., [tex]\( x \)[/tex]) must also be non-negative: [tex]\( x \ge 0 \)[/tex].
- If [tex]\( m > 0 \)[/tex], [tex]\( g(x) = m\sqrt{x} \)[/tex] will yield non-negative values, specifically any non-negative real number as [tex]\( x \)[/tex] varies over all non-negative values.
- If [tex]\( m < 0 \)[/tex], [tex]\( g(x) = m\sqrt{x} \)[/tex] will yield non-positive values, specifically any non-positive real number because [tex]\( \sqrt{x} \ge 0 \)[/tex] for [tex]\( x \ge 0 \)[/tex].

### Step 3: Compare the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
- For the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to be the same:
- If [tex]\( m > 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] and [tex]\( g(x) = m\sqrt{x} \)[/tex] both yield any non-negative real number.
- If [tex]\( m < 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] yields any non-negative real number, but [tex]\( g(x) = m\sqrt{x} \)[/tex] yields any non-positive real number.

### Conclusion:
The ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] can only be the same when [tex]\( m > 0 \)[/tex].

Thus, the statement that is true about the value of [tex]\( m \)[/tex] is:

[tex]\[ m \text{ can be any positive real number.} \][/tex]
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