Chapter 6: Problem 2

State the rule of each of the following functions: (a) \(f(x)=5 x\) (b) \(f(t)=5 t\) (c) \(f(x)=8 x+10\) (d) \(f(t)=7 t-27\) (e) \(f(t)=1-t\) (f) \(h(t)=\frac{t}{3}+\frac{2}{3}\) (g) \(f(x)=\frac{1}{1+x}\)

Short Answer

Expert verified

Answer: The rule of the function \(f(x)=\frac{1}{1+x}\) is to add \(1\) to the input value and then take the reciprocal of the sum.

Step by step solution

(a) Function \(f(x)=5x\) Rule

This function is already in the form \(f(x)=mx\), where \(m\) is the slope. The slope of this function is \(m=5\). Therefore, the rule of the function \(f(x)=5x\) is to multiply the input value by \(5\).

(b) Function \(f(t)=5t\) Rule

Similarly to (a), this function is in the form \(f(t)=mt\), with \(m=5\). The rule of the function \(f(t)=5t\) is to multiply the input value by \(5\).

(c) Function \(f(x)=8x+10\) Rule

This function is in the form \(f(x)=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope of this function is \(m=8\) and the y-intercept is \(b=10\). The rule of this function is to multiply the input value by \(8\) and then add \(10\).

(d) Function \(f(t)=7t-27\) Rule

This function is in the form \(f(t)=mt+b\), with the slope \(m=7\) and the y-intercept \(b=-27\). The rule of the function \(f(t)=7t-27\) is to multiply the input value by \(7\) and then subtract \(27\).

(e) Function \(f(t)=1-t\) Rule

This function is in the form \(f(t)=1-mt\), where the slope is, in this case, \(m=-1\). The rule of the function \(f(t)=1-t\) is to subtract the input value from \(1\).

(f) Function \(h(t)=\frac{t}{3}+\frac{2}{3}\) Rule

This function is in the form \(f(t)=\frac{mt}{n}+\frac{b}{n}\), with the slope \(m=1\) and the y-intercept \(b=2\). The rule of the function \(h(t)=\frac{t}{3}+\frac{2}{3}\) is to divide the input value by \(3\) and then add \(\frac{2}{3}\).

(g) Function \(f(x)=\frac{1}{1+x}\) Rule

We cannot represent this function in the same linear form as the previous examples, but we can still describe its rule so that inputting the values will be easy to understand. The rule of the function \(f(x)=\frac{1}{1+x}\) is to add \(1\) to the input value and then take the reciprocal of the sum.

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State the rule of each of the following functions: (a) \(f(x)=5 x\) (b) \(f(t)=5 t\) (c) \(f(x)=8 x+10\) (d) \(f(t)=7 t-27\) (e) \(f(t)=1-t\) (f) \(h(t)=\frac{t}{3}+\frac{2}{3}\) (g) \(f(x)=\frac{1}{1+x}\)

Short Answer

Step by step solution

(a) Function \(f(x)=5x\) Rule

(b) Function \(f(t)=5t\) Rule

(c) Function \(f(x)=8x+10\) Rule

(d) Function \(f(t)=7t-27\) Rule

(e) Function \(f(t)=1-t\) Rule

(f) Function \(h(t)=\frac{t}{3}+\frac{2}{3}\) Rule

(g) Function \(f(x)=\frac{1}{1+x}\) Rule

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