SOLUTION
Given the question in the image, the following are the solution steps to get the answer
Step 1: Define antiderivative
Antiderivative of a function is nothing but integral with respect to x
[tex]F(x)=\int f(x)dx[/tex]Step 2: Find the antiderivative of the given function in the question
[tex]\begin{gathered} f(x)=12x-9 \\ F(x)=\int (12x-9)dx \\ \text{Providing integral to each term, we have} \\ F(x)=\int 12xdx-\int 9dx \\ F(x)=12\int xdx-9\int x^0dx---(1) \\ we\text{ know that} \\ \int x^ndx=\frac{x^{^{n+1}^{}}}{n+1},n\ne1 \\ F(x)=12\times\frac{x^{1+1}}{1+1}-9\times\frac{x^{0+1}}{0+1}+c \\ F(x)=12\times\frac{x^2}{2}-9\times\frac{x}{1}+c \\ F(x)=6x^2-9x+c----(2) \end{gathered}[/tex]Step 3: we find the value of c in front of the integration formula
[tex]\begin{gathered} \text{Given that F(1)=7} \\ 6(1)^3-9(1)+c=7 \\ 6-9+c=7 \\ c=7-6+9 \\ c=10 \end{gathered}[/tex]Step 4: We write the final result for F(x)
[tex]\begin{gathered} \text{Hence, we have:} \\ F(x)=6x^2-9x+10 \end{gathered}[/tex]Hence, the final result for F(x) is given as:
[tex]F(x)=6x^2-9x+10[/tex]