Question

Which function would horizontally stretch r(x)=−2x−−√−5 by a factor of 3?(1 point) Responses m(x)=−22x3−−√−5 m left parenthesis x right parenthesis equals negative 2 Start Root Start Fraction 2 x over 3 End Fraction End Root minus 5 z(x)=−23x2−−√−5 z left parenthesis x right parenthesis equals negative 2 Start Root Start Fraction 3 x over 2 End Fraction End Root minus 5 d(x)=−2x3−−√−5 d left parenthesis x right parenthesis equals negative 2 Start Root Start Fraction x over 3 End Fraction End Root minus 5 b(x)=−23x−−√−5

Answer 1

To horizontally stretch a function by a factor of 3, we will replace \( x \) with \( \frac{x}{3} \).

The original function is:
\[ r(x) = -2\sqrt{x} - 5 \]

To stretch this function horizontally by a factor of 3, we will use:
\[ m(x) = r\left(\frac{x}{3}\right) \]

Now, substituting into the original function:
\[ m(x) = -2\sqrt{\frac{x}{3}} - 5 \]

Among the options given, let's analyze each one:

- \( m(x) = -2\sqrt{\frac{2x}{3}} - 5 \) (not a correct stretch)
- \( z(x) = -\frac{2}{3}\sqrt{\frac{3x}{2}} - 5 \) (not a correct stretch)
- \( d(x) = -2\sqrt{\frac{x}{3}} - 5 \) (this matches our derived function)
- \( b(x) = -\frac{2}{3}\sqrt{x} - 5 \) (not a correct stretch)

The correct response is:
\[ d(x) = -2\sqrt{\frac{x}{3}} - 5 \]