Question

your functions
here are descriptions and equations that represent four functions.
a ( f(x) = 3x - 7 )
b ( g(x) = 3(x - 7) )
c ( h(x) = \frac{x}{3} - 7 )
d ( k(x) = \frac{x - 7}{3} )
1 to get the output, subtract 7 from the input, then divide the result by 3.
2 to get the output, subtract 7 from the input, then multiply the result by 3.
3 to get the output, multiply the input by 3, then subtract 7 from the result.
4 to get the output, divide the input by 3, then subtract 7 from the result.
1 match each equation with a verbal description that represents the same function. record your results.
2 for one of the functions, when the input is 6, the output is -3. which is that function: ( f, g, h, ) or ( k )?
explain how you know.
3 which of the four functions have the greatest value when the input is 0? what about when the input is 10?
are you ready for more?
mai says ( f(x) ) is always greater than ( g(x) ) for the same value of ( x ). is this true?
explain how you know.

Explanation:

Sub - Question 1: Matching Functions with Descriptions

Step 1: Analyze \( f(x)=3x - 7\)

The function \( f(x)=3x - 7\) means we first multiply the input \( x\) by 3 (i.e., \( 3x\)) and then subtract 7. So it matches description 3.

Step 2: Analyze \( g(x)=3(x - 7)\)

For \( g(x)=3(x - 7)\), we first subtract 7 from the input \( x\) (i.e., \( x - 7\)) and then multiply the result by 3. So it matches description 2.

Step 3: Analyze \( h(x)=\frac{x}{3}-7\)

The function \( h(x)=\frac{x}{3}-7\) means we first divide the input \( x\) by 3 (i.e., \( \frac{x}{3}\)) and then subtract 7. So it matches description 4.

Step 4: Analyze \( k(x)=\frac{x - 7}{3}\)

For \( k(x)=\frac{x - 7}{3}\), we first subtract 7 from the input \( x\) (i.e., \( x - 7\)) and then divide the result by 3. So it matches description 1.

Step 1: Evaluate \( f(6)\)

For \( f(x)=3x - 7\), substitute \( x = 6\): \( f(6)=3\times6-7=18 - 7 = 11\)

Step 2: Evaluate \( g(6)\)

For \( g(x)=3(x - 7)\), substitute \( x = 6\): \( g(6)=3\times(6 - 7)=3\times(-1)=-3\)

Step 3: Evaluate \( h(6)\)

For \( h(x)=\frac{x}{3}-7\), substitute \( x = 6\): \( h(6)=\frac{6}{3}-7 = 2-7=-5\)

Step 4: Evaluate \( k(6)\)

For \( k(x)=\frac{x - 7}{3}\), substitute \( x = 6\): \( k(6)=\frac{6 - 7}{3}=\frac{-1}{3}\approx - 0.33\)

Step 1: Evaluate functions at \( x = 0\)

• For \( f(0)=3\times0-7=-7\)
• For \( g(0)=3\times(0 - 7)=3\times(-7)=-21\)
• For \( h(0)=\frac{0}{3}-7=0 - 7=-7\)
• For \( k(0)=\frac{0 - 7}{3}=\frac{-7}{3}\approx - 2.33\)

Among \(-7\), \(-21\), \(-7\), and \(-2.33\), the greatest value is \(-2.33\) (from \( k(x)\)).

Step 2: Evaluate functions at \( x = 10\)

• For \( f(10)=3\times10-7=30 - 7 = 23\)
• For \( g(10)=3\times(10 - 7)=3\times3 = 9\)
• For \( h(10)=\frac{10}{3}-7=\frac{10 - 21}{3}=\frac{-11}{3}\approx - 3.67\)
• For \( k(10)=\frac{10 - 7}{3}=\frac{3}{3}=1\)

Among \(23\), \(9\), \(-3.67\), and \(1\), the greatest value is \(23\) (from \( f(x)\)).

Answer:

A - 3, B - 2, C - 4, D - 1

Sub - Question 2: Finding the Function with Input 6 and Output - 3