The function 𝑓f is defined by 𝑓(𝑥)=2𝑥+147f(x)= 72x+14​ , and 𝑓(𝑎)=6f(a)=6, where 𝑎a is a constant. What is the value of 𝑎a?

1. Break Down the Problem

We are given the function $f(x) = 72x + 14$ and the condition $f(a) = 6$. We need to find the value of the constant $a$.

2. Relevant Concepts

We will use the function definition to set up an equation that we can solve for $a$: $f(a) = 72a + 14$

3. Analysis and Detail

Using the information provided, we can substitute into the equation: $72a + 14 = 6$

Now, we will solve for $a$:

1. Subtract 14 from both sides. $72a = 6 - 14$ $72a = -8$

2. Divide both sides by 72. $a = \frac{-8}{72}$ $a = \frac{-1}{9}$

4. Verify and Summarize

To verify, we can substitute $a = \frac{-1}{9}$ back into the function to check if $f(a) = 6$: $f\left(\frac{-1}{9}\right) = 72\left(\frac{-1}{9}\right) + 14 = -8 + 14 = 6$ The calculation is correct.

Final Answer

The value of $a$ is $\frac{-1}{9}$.