for the function f(x)=6-5(3)^x-1/3
(a)simplify(in form =ab^kx+d
(b)state the domain
(c)state the range
To simplify the function f(x) = 6 - 5(3)^(x - 1/3) and express it in the form ab^kx + d, we can follow these steps:
Step 1: Simplify the exponent:
The exponent x - 1/3 can be written as (3^(1/3))^(-1/3). Applying the rule of exponents (a^m)^n = a^(m*n), we can simplify it to 3^((1/3) * (-1/3)), which equals 3^(-1/9).
Step 2: Substitute the simplified exponent in the given function:
Substituting 3^(-1/9) back into the function, we get:
f(x) = 6 - 5 * 3^(-1/9)
Step 3: Write it in the desired form:
To express the function in the form ab^kx + d, we need to rewrite it further. To do that, let's rationalize the denominator of the exponent.
Step 4: Rationalize the denominator:
We can rationalize the denominator by multiplying the numerator and denominator of the exponent by the conjugate of the denominator (3^(-1/9) + 1). The conjugate of (3^(-1/9) + 1) is (3^(-1/9) + 1), so the rationalization is as follows:
f(x) = 6 - 5 * 3^(-1/9) * (3^(-1/9) + 1) / (3^(-1/9) + 1)
Step 5: Simplify the expression:
Expanding the numerator and denominator, we get:
f(x) = 6 - 5 * (3^(-2/9) + 3^(-1/9)) / (3^(-1/9) + 1)
Step 6: Combine like terms:
Now, we can simplify the expression further by combining like terms:
f(x) = 6 - 5 * (1/3^2 + 1/3^1) / (1/3^1 + 1)
Step 7: Evaluate the exponents:
Calculating the exponents, we have:
f(x) = 6 - 5 * (1/9 + 1/3) / (1/3 + 1)
Step 8: Combine fractions and simplify:
Combining the fractions in the numerator and denominator, we get:
f(x) = 6 - 5 * (4/9) / (4/3)
Step 9: Simplify and write in desired form:
To simplify further, we can cancel out the common factor of 4 from the numerator and denominator, resulting in:
f(x) = 6 - 5 * (1/9) / (1/3)
Simplifying the expression, we have:
f(x) = 6 - (5/9) / (1/3)
Finally, by dividing the numerator by the denominator, we get:
f(x) = 6 - (5/9) * (3/1)
Simplifying further, we have:
f(x) = 6 - (15/9)
Combining like terms, we get:
f(x) = (54 - 15)/9
Therefore, the simplified form of f(x) in the form ab^kx + d is:
f(x) = (39/9), which can be simplified as f(x) = 13/3.